GIFT   OF 


S>emuCentennial 


ELIZABETHAN  TRANSLATIONS  FROM  THE  ITAL- 
IAN. By  MARY  AUGUSTA  SCOTT,  Ph.D.  (A.B.  Vas- 
sar, 1876),  Professor  of  English  Literature  in  Smith 
College. 

SOCIAL  STUDIES  IN  ENGLISH  LITERATURE. 
By  LAURA  J.  WVLIE,  Ph.D.  (A.B.  Vassar,  1877),  Pro- 
fessor of  English  in  Vassar  College. 

THE  LEARNED  LADY  IN  THE  EIGHTEENTH 
CENTURY.  By  MVRA  REYNOLDS,  Ph.D.  (A.B.  Vas- 
sar, 1880),  Professor  of  English  Literature  in  Chicago 
University.  [/« preparation] 

THE  CUSTOM  OF  DRAMATIC  ENTERTAINMENT  IN 
SHAKESPEARE'S  PLAYS.  By  ORIH  J.  HATCHER, 
Ph.D.  (A.B.  Vassar,  1888),  Formerly  Associate  Pro- 
fessor of  Comparative  Literature  in  Bryn  Mawr  Col- 
lege. {In  preparation.] 

INTRODUCTION  TO  THE  STUDY  OF  VARIABLE 
STARS.  By  CAROLINE  E.  FURNBSS,  Ph.D.  (A.B.  Vas- 
sar, 1891),  Professor  of  Astronomy  in  Vassar  College. 

MOVEMENT  AND  MENTAL  IMAGERY.  By  MAR- 
GARET FLOY  WASHBURN,  Ph.D.  (A.B.  Vassar,  1891), 
'  Professor  of  Psychology  in  Vassar  College.  [/» prep- 
aration^ 

BRISSOT  DE  WARVILLE:  A  STUDY  IN  THE  HIS- 
TORY OF  THE  FRENCH  REVOLUTION.  By  ELOISE 
ELLERY,  Ph.D.  (A.B.  Vassar,  1897),  Associate  Profes- 
sor of  History  in  Vassar  College. 

HOUGHTON  MIFFLIN  COMPANY 
BOSTON  AND  NEW  YORK 


AN  INTRODUCTION  TO  THE   STUDY 
OF    VARIABLE  STARS 


Plate  I 

ATLAS  WITH  THE  GLOBE.  NAPLES  MUSEUM 


«&emi-€entenmai 


AN  INTRODUCTION  TO 

THE  STUDY 
OF  VARIABLE  STARS 


BY 

CAKOLINE  E.  FUKNESS,  PH.D. 

Director  of  the  Vassar  College  Observatory 


With  Illustrations 


BOSTON  AND  NEW  YORK 
HOUGHTON  MIFFLIN  COMPANY 

re?s  Cambridge 
1915 


COPYRIGHT,   1915,  BY  CAROLINE  E.   FURNESS 
ALL  RIGHTS  RESERVED 

Published  October  1315 


PUBLISHED  IN  HONOR   OF  THE 
FIFTIETH  ANNIVERSARY 

OF  THE 

FOUNDING  OF  VASSAR  COLLEGE 
1865-1915 


DEDICATED  TO 

MARY  W.  WHITNEY 

IN  MEMORY   OF  MANY  HAPPY  HOURS 

SPENT  TOGETHER  IN  THE  PURSUIT  OF 

OUR  LOVED  SCIENCE 


PREFACE 

DURING  the  past  few  years  the  subject  of  variable  stars 
has  become  increasingly  interesting  to  the  amateur  who  is  the 
owner  of  a  telescope,  as  well  as  to  the  average  college  student 
who  has  some  knowledge  of  astronomy,  while  to  the  research 
worker  it  offers  many  lines  of  investigation  which  are  full  of 
promise.  However,  so  complex  is  the  subject,  and  so  diverse 
the  principles  involved  in  a  complete  understanding  of  it,  that 
extensive  reading  in  several  different  directions  is  required  as 
a  foundation. 

It  is  with  the  purpose  of  supplying  this  need  as  well  as  of 
making  an  important  and  attractive  branch  of  astronomy 
accessible  to  the  student  that  the  present  volume  has  been 
prepared.  It  is  the  outcome  of  several  years  of  teaching  the 
subject  in  Vassar  College,  for  which  the  material  was  prima- 
rily collected.  This  material  is  scattered  throughout  various 
periodicals  in  the  form  either  of  research  papers  or  quite  popu- 
lar articles,  intended  to  give  directions  for  observation  to 
owners  of  small  telescopes.  A  large  amount  of  historical  matter 
is  also  included,  which  is  taken  from  sources  not  within  easy 
reach  of  the  general  reader.  Mention  may  be  made  of  some  of 
the  subjects  treated,  which  are  introductory  to  the  study  of 
stellar  variation,  such  as  the  study  of  the  Durchmusterung 
charts,  photometry  in  all  its  branches,  spectroscopy,  and  star 
color.  The  purpose  of  the  present  volume  is  to  consider  all  of 
these  points,  and  in  particular  to  give  in  as  simple  and  clear  a 
form  as  possible  a  full  presentation  of  the  physical  principles 
upon  which  many  of  the  instruments  and  methods  of  investi- 
gation are  based,  principles  such  as  those  of  polarized  light, 
spectrum  analysis,  the  formation  of  the  photographic  image, 
and  photo-electricity.  Textbooks  on  astronomy  rarely  include 


x  PREFACE 

such  preliminary  matters,  even  though  they  are  not  subjects 
which  the  student  is  necessarily  expected  to  know. 

Thus  far  no  general  book  on  variable  stars  has  been  pre- 
sented to  the  public  in  English.  In  German  a  comprehensive 
treatise  is  being  prepared  by  Father  Hagen,  and  issued  a  sec- 
tion at  a  time.  Two  parts  have  already  appeared,  and  in  the 
introduction  to  the  first,  which  is  called  the  historical  techni- 
cal part,  Hagen  states  that  it  is  primarily  a  collection  of  sources, 
and  that  brief  handbooks  in  different  languages  can  easily  be 
formed  from  the  material  included.  The  author,  in  a  personal 
interview  with  this  distinguished  astronomer  at  Rome,  in  1914, 
received  encouragement  from  him  to  proceed  with  her  project, 
and  permission  to  use  any  of  the  material  in  his  treatise.  The 
present  volume,  however,  is  an  introduction  rather  than  a  hand- 
book, and  as  such  devotes  more  space  to  explanatory  mate- 
rial than  to  an  extensive  treatment  of  the  results  of  research. 
Much  of  the  material  had  already  been  collected  before  1914, 
but  frequent  reference  to  Hagen's  work  will  be  found  in  the 
footnotes. 

[  The  writer  wishes  now  to  express  her  indebtedness  to  her 
many  friends  who  have  assisted  her  at  various  points  in  this 
undertaking;  first  to  her  astronomical  colleagues,  who  looked 
over  the  outline  and  made  valuable  suggestions  as  to  the  points 
which  should  be  included  in  it;  to  Professors  Schlesinger  and 
Jordan,  of  the  Allegheny  Observatory;  to  Professors  Frost  and 
Parkhurst,  of  Yerkes;  and  to  Professor  Pickering,  of  the  Har- 
vard Observatory.  It  is  owing  to  suggestions  from  these  as- 
tronomers that  the  chapter  on  photo-electric  cells  was  included, 
and  that  so  much  space  was  given  to  photographic  photome- 
try and  star  colors.  They  also  freely  offered  the  use  of  any 
material  from  their  publications  which  might  be  desired. 

Whatever  clearness  of  presentation  there  may  be  in  the  dis- 
cussion of  the  photo-electric  cell  the  writer  owes  to  her  col- 
league at  Vassar,  Professor  Saunders,  of  the  Physics  Depart- 
ment, who  gave  generously  of  his  time  to  the  discussion  of  that 
difficult  and  unfamiliar  subject,  as  well  as  of  several  other 


PREFACE  xi 

technical  points.  Miss  Ernestine  Fuller,  of  the  Astronomical 
Department,  assisted  by  looking  up  references  at  several 
points  and  criticized  the  presentation  of  some  of  the  physical 
principles.  Professor  Treadwell  has  given  useful  suggestions 
in  regard  to  the  drawings. 

At  the  suggestion  of  Miss  Helen  Swartz  several  items  were 
included  which  were  thought  to  be  useful  to  the  non-profes- 
sional observer.  She  also  read  a  large  part  of  the  manuscript 
and  made  valuable  criticisms  of  the  form. 

The  writer  wishes  to  express  her  thanks  also  to  her  students 
in  the  course  on  variable  stars  during  the  present  year,  Miss 
Vera  Ringwood  and  Miss  Evelyn  Wickham,  with  whom  she 
has  held  many  discussions  as  to  the  form  of  presentation,  and 
who,  by  their  interest  and  candid  criticism,  have  aided  greatly 
in  maintaining  the  standard  of  clearness  which  she  has  striven 
to  reach. 

Mr.  Olcott,  secretary  of  the  American  Variable  Star  Section, 
made  suggestions  as  to  what  points  the  amateur  observer 
would  be  especially  interested  in,  and  sent  several  items  which 
have  been  incorporated  in  the  chapter  on  "Hints  to  Observ- 
ers." Mr.  David  Blencoe,  also  a  member  of  the  Association, 
has  kindly  sent  his  work  on  a  statistical  study  of  variable  stars, 
which  had  been  prepared  for  private  circulation. 

The  author's  greatest  debt,  however,  is  to  Miss  Helen  Van 
Kleeck,  who  at  several  times  in  the  past  has  assisted  in  prepar- 
ing the  publications  of  Vassar  College  for  the  press.  To  her 
faithful  and  intelligent  work  in  transcription  the  writer  owes 
the  completion  of  the  volume  in  the  required  time,  and  to  her 
careful  criticism  is  due  much  of  the  clearness  of  style.  Miss 
Van  Kleeck  also  prepared  the  drawings  for  the  illustrations, 
which  are  provided  for  by  the  publication  fund  of  the  Observa- 
tory. 

The  observation  of  variable  stars  was  introduced  into  the 
program  of  the  Vassar  Observatory  by  Professor  Mary  W. 
Whitney  in  1901,  and  when  later  the  subject  was  made  a  regu- 
lar course  of  study  in  the  astronomical  department,  the  writer 


xii  PREFACE 

co-operated  with  her  for  the  first  few  years  in  giving  the  in- 
struction. The  writer  cannot  adequately  express  her  constant 
indebtedness  to  Professor  Whitney  for  the  opportunity  and 
encouragement  afforded  her  during  all  her  years  of  work  at 
Vassar. 

CAROLINE  E.  FUBNESS. 

VASSAB  COLLEGE  OBSERVATORY, 
March  26,  1915. 


CONTENTS 

I.  INTRODUCTORY 

General  description  of  stellar  variation.  Elements  of 
variation.  Julian  Day.  Classification  of  variables.  Prin- 
ciples of  spectrum  analysis.  Classification  of  stellar  spec- 
tra. Connection  between  spectral  type  and  type  of  varia- 
tion   3 

H.  STAR  CHARTS  FOR  GENERAL  USE 

Argelander,  Uranometria  Nova.  Heis,  Atlas  Coelestis. 
Schurig,  Himmels  Atlas.  Upton,  Klein,  etc.  Argelander, 
Banner  Durchrmisterung,  detailed  account,  with  sugges- 
tions for  its  use.  Palisa  and  Wolf,  photographic  maps. 
Ecliptic  charts.  Charts  of  Carte  du  del.  Precession  .  38 

HI.  STAR  CHARTS  FOR  VARIABLES 

Hagen,  Atlas  Stellarum  Variabilium,  very  detailed  ac- 
count, with  directions  for  its  use.  Parkhurst,  enlarge- 
ments. Harvard  photographic  maps.  Miscellaneous  .  52 

IV.  CATALOGUES  OF  VARIABLES 

Naming  of  variables.  Harvard,  Hartwig,  Chandler. 
Methods  of  discovery;  visual,  photographic,  spectro- 
scopic.  Illustrations.  Systematic  search  ....  68 

V.  STELLAR  MAGNITUDE 

Early  history.  Ptolemy,  Bayer,  and  others,  before  the 
invention  of  the  telescope.  Herschel.  Argelander,  mag- 
nitudes in  the  Durchmusterung.  Gould.  Pogson's  rule. 
Fechner 81 

VI.  VISUAL  PHOTOMETRY 

Methods  of  observing  variable  stars  visually.  Arge- 
lander's  step  method.  Harvard  method  of  relative  bright- 
ness. Use  of  direct  magnitude.  Precautions  in  observing. 


xiv  CONTENTS 

Purkinje  phenomenon.  Dove's  phenomenon.  Polarized 
light.  Zb'llner  photometer.  Harvard  meridian  photome- 
ter. Pickering's  wedge  photometer 103 

VH.  PHOTOGRAPHIC  PHOTOMETRY 

The  formation  of  star  images.  Chromatic  aberration 
and  color  curve.  Visual  and  photographic  telescopes. 
Effect  of  color  on  size  of  image.  Formulas  for  deriving 
magnitudes  from  measurements  of  photographic  images. 
Stained  plates.  Connection  with  spectral  type.  Extra- 
focal  images.  Comparisons  by  Argelander's  method. 
Work  at  Harvard 130 

VHI.  PHOTO-ELECTRIC  PHOTOMETRY 

Work  of  Stebbins  with  selenium  cell.  Principles  of 
photo-electric  action.  Electrometer.  Work  at  Berlin- 
Babelsberg.  Description  of  instrument  and  the  results. 
General  conclusions  .  .  .  .  .  .  .  .  .154 

IX.  FORMATION  OF  LIGHT  SCALE 

Combination  of  observations  made  by  the  Argelander 
method.  Formation  of  light  scale  for  comparison  stars 
without  weights.  Light  scale  for  variables.  Light  step 
converted  into  magnitude.  Plotting  of  light  curve  by 
light  step  or  magnitude.  Determination  of  maximum  or 
minimum  from  single  light  curve.  Use  of  weights  in 
forming  scale 170 

X.  MEAN  LIGHT  CURVE 

Use  of  Heis's  observations  of  8  Cephei.  Determina- 
tion of  period  and  epoch  from  light  curve.  Correction  of 
approximate  elements  by  later  observations.  Mean 
light  curve.  Use  of  mean  light  curve  in  determining  min- 
imum of  Algol  type.  Harvard  method  for  long  period 
variables .  ...  .  .186 

XI.  PREDICTION  OF  MAXIMA  AND  MINIMA  FROM 
THE  ELEMENTS 

Explanation  of  Julian  Day.  Simple  formula.  Addi- 
tion of  periodic  term.  Secular  term.  Reduction  to  sun 
for  short  period  variables 204 


CONTENTS  xv 

XII.  ECLIPSING  BINARIES 

Doppler's  principle.  Spectrographic  apparatus.  Ap- 
plication to  variable  stars.  Explanation  of  Algol  type. 
Evidence  from  light  curve.  Spectroscopic  evidence. 
General  relations,  ft  Lyrae  type.  Cepheid  variable 
type.  Cluster  type 216 

XHI.  LONG  PERIOD  VARIABLES 

Irregular  variables.  Temporary  stars.  Collections  of 
observations:  Argelander,  Schb'nfeld,  Heis,  Krueger, 
Schmidt,  Pogson,  Knott,  Goodricke  and  Pigott,  Gould, 
Chandler 248 

XIV.  STATISTICAL  STUDY 

Correlation  of  number,  length  of  period,  color,  range 
of  brightness,  and  spectral  type  for  long  and  short  period 
variables.  Galactic  distribution.  Miscellaneous  facts  .  273 

XV.  HINTS  FOR  OBSERVERS 

Circulars  issued  by  Harvard  College  Observatory  in- 
viting co-operation.  American  Association  of  Variable 
Star  Observers.  British  Association.  Hints  for  observ- 
ers: (1)  Use  of  telescope;  (2)  Time;  (3)  Identification  of 
variable;  (4)  Methods  of  recording  ;  (5)  Precautions. 
Stars  suitable  for  observers  with  telescopes  of  different 
aperture,  mounted  or  unmounted.  Brief  bibliography.  .  289 

APPENDIX 

Table   I.  Julian  Days 311 

Table  II.  Fractions  of  a  day  .      .      .      .      .      .      .312 

Explanation  of  Tables 313 

/  Description  of  plates  of  stellar  spectra 315 

INDEX  .  319 


PLATES 

I.  Atlas  with  the  Globe.  Naples  Museum    .      .      .  Frontispiece 
H.  Typical  Stellar  Spectra   .      .      r     .      .      ....      .33 

III.  Frontispiece  from  Bayer's  Uranometria,  1639        ...     83 

IV.  The  Constellation  Gemini  from  Bayer's  Uranometria  .       .     85 

V.  Double-slide  plate-carrier  on  the  40-inch  Telescope.  Yerkes 

Observatory 139 

VI.  The  two-foot  Reflector.  Yerkes  Observatory        .       .       .141 
VII.  The  Bruce  Photographic  Telescope.  Yerkes  Observatory  .  145 

VIII.  The  Bruce  Spectrograph  of  the  Yerkes  Observatory,  at- 
tached to  the  Telescope     219 

IX.  The  Bruce  Spectrograph  of  the  Yerkes  Observatory,  ul- 
terior structure 223 

X.  Spectra  of   £  Ursae  Majoris,  showing  lines   single  and 

double 235 

XI.  Spectra  of  //,  Orionis,  showing  different  amounts  of  radial 

velocity 237 

XII.  Spectra  of  Sun  and  Type  A 318 

XIII.  Typical  Spectra 318 

XIV.  Peculiar  Spectra.  318 


, 


FIGURES  IN  THE  TEXT 

1.  Single  Light  Curve  of  S  Ursae  Majoris        .      ....  4 

2.  Light  Curve  of  Nova  Persei     .      .      .      .      .      .      .      .  7 

3.  Mean  Light  Curve  of  T  Cassiopeiae     .      .,*..& 

4.  Mean  Light  Curve  of  R  Ursae  Majoris       ...      ,      .  0 

5.  Light  Changes  of  SS  Cygni       .      .       .      .....  10 

6.  Mean  Light  Curve  of  8  Cephei 11 

7.  Mean  Light  Curve  of  S  Arae 12 

8.  Mean  Light  Curve  of  /?  Lyrae 13 

9.  Light  Curve  of  £  Geminorum 14 

10.  Light  Curve  of  Algol 15 

11.  Mean  Light  Curve  of  U  Cephei      .      .      .      .      .      .      .16 

12.  Prism  Spectroscope .       .       .       .18 

13.  Diagram  of  a  Complete  Wave 20 

14.  The  Solar  Spectrum 28 

15.  Square  from  Durchmusterung  Chart 47 

16.  Magnitude  Curve  for  RV  Hydrae 59 

17.  The  Nicol's  Prism .      .  115 

18.  The  Zollner  Photometer 117 

19.  Pickering's  Meridian  Photometer 123 

20.  Wedge  Photometer,  Yerkes  Observatory 127 

21.  Color  Curve  of  the  40-inch  Objective.  Yerkes  Observatory .  143 

22.  Spectral  Type  and  Color  Intensity 150 

23.  Diagram  of  the  Photo- Electric  Apparatus.   Berlin-Babels- 

berg 161 

24.  Single  Light  Curves  of  8  Cephei 175 

25.  Light  Curve  of  o  Ceti 178 

26.  Magnitude  Curve  for  o  Ceti 180 


xx  FIGURES  IN   THE  TEXT 

27.  Mean  Light  Curve  of  3  Cephei 199 

28.  Diagram  for  obtaining  the  Reduction  to  the  Sun     .       .       .  214 

29.  Diagram  I.  The  Light  Curve  of  an  Eclipsing  Binary     .       .  231 

30.  Diagram  II.  The  Relative  Orbit 232 

31.  Diagram  HI.  The  Real  Orbits 233 

32.  Diagram  IV.  Spectroscopic  Evidence 234 

33.  Diagram  V.  The  Velocity  Curve 234 

34.  The  System  of  /?  Lyrae 242 

35.  Theoretical  System  of  8  Cephei 247 

36.  Relation  between  Color  and  decreasing  Brightness  .       .       .  277 


AN  INTRODUCTION  TO  THE   STUDY 
OF   VARIABLE  STARS 


ABBREVIATIONS 

Annals,  H.C.O.,  Annals  of  the  Harvard  College  Observatory. 

A.  G.,  Astronomische  Gesellschaft. 

A.  N.,  Astronomische  Nachrichten. 

Ast.  Jour.,  Astronomical  Journal. 

Ap.  J.,  Astrophysical  Journal. 

H.C.O.  Circ.,  Harvard  College  Observatory  Circulars. 

L.O.B.,  Lick  Observatory  Bulletins. 

Mem.  R.A.S.,  Memoirs  of  the  Royal  Astronomical  Society. 

Phil.  Trans.,  Philosophical  Transactions  of  the  Royal  So- 

ciety, London. 

Physik.  Zeits.,  Physikalische  Zeitschrift. 

Pop.  Ast.,  Popular  Astronomy. 

Potsdam  Phot.  DM.,  Potsdam  Photometric  Durchmusterung. 
Rad.  Obs.,  Radcliffe  Observatory  Publications. 

Uran.  Arg.,  Uranometria  Argentina. 

Ver.  St.,  I  Veranderliche  Sterne,  by  Hagen. 

V.J.S.,  VVierteljahrsschrift  der  Astronomischen  Gesell- 

v  schaft. 


AN  INTRODUCTION  TO  THE  STUDY  OF 
VARIABLE  STARS 

CHAPTER  I 

INTRODUCTORY 

WE  shall  take  it  for  granted  that  the  reader  is  already  ac- 
quainted with  the  main  facts  of  Astronomy,  but  since  this 
does  not  necessarily  include  a  knowledge  of  the  points  which 
bear  directly  upon  the  study  of  variable  stars,  a  brief  resume 
of  them  will  be  given  in  this  introductory  chapter.  However, 
the  statements  made  here  are  to  be  considered  as  preliminary 
only,  and  each  will  be  more  fully  discussed  in  some  later  chap- 
ter. The  topics  presented  will  be  a  general  description  of  stel- 
lar variation,  the  elements  of  variation,  classes  of  variables, 
the  general  principles  underlying  spectrum  analysis,  the  classi- 
fication of  stellar  spectra,  and  the  connection  between  the 
spectral  type  and  the  type  of  variation. 

GENERAL  DESCRIPTION   OF   STELLAR  VARIATION 

A  variable  star  is  one  that  undergoes  a  change  in  brightness. 
With  some  stars  the  change  is  as  great  as  four  or  even  six  mag- 
nitudes, while  with  others  it  may  be  only  one  magnitude,  and 
in  some  cases  as  small  as  half  a  magnitude.  This  change  in 
brightness  is  observed  by  comparing  the  light  of  the  variable 
with  the  light  of  some  standard  star  which  is  assumed  to  be 
constant  in  brightness,  the  comparison  being  made  either 
directly,  or  through  the  medium  of  some  sort  of  artificial  star. 
The  different  methods  of  making  the  comparisons  will  be  dis- 
cussed at  length  in  the  chapters  on  photometry.  It  is  sufficient 
here  to  state  that  the  result  of  the  observations  is  to  furnish 
the  magnitudes  of  the  variables  at  certain  recorded  instants 


STITDY  OF  VARIABLE  STARS 

of  time.  In  order  to  represent  the  variation  to  the  eye,  the 
data  are  plotted  on  co-ordinate  paper,  using  the  time  as  the 
horizontal  co-ordinate,  or  abscissa,  and  the  observed  magni- 
tude as  the  vertical  co-ordinate  or  ordinate.  A  smooth  curve 
is  then  drawn  through  the  points  which  is  called  the  single 
light  curve  of  the  variable.  This  is  illustrated  in  the  following 
diagram. 


7.0 
8.0 
9.0 
10.0 
11.0 

JLD 
14 

STc 

/•yto 

K 

\\J 

c 

A 

) 

\ 

f 

\ 

1 

V 

i 

£ 

<i 

\^  ^ 

3 

1 

\ 

V^ 

3 

N 

y 

5 

( 

I 

\ 

c/ 

\j 

¥ 

c 

12,900             3000                    3100                    32.00 

Figure  1 

SINGLE  LIGHT  CURVE  OF  S  URSAE  MAJORIS 
ELEMENTS   OF  VARIATION 

When  a  long  series  of  observations  has  been  made  and  the 
results  plotted  as  just  described,  a  study  of  the  curves  will 
show  that  the  same  form  recurs  with  more  or  less  regularity, 
and  that  certain  quantities  can  be  determined  which  will  de- 
scribe it.  They  are  the  magnitude  at  maximum,  the  magnitude 
at  minimum,  and  the  length  of  the  period,  i.e.,  the  time  from 
one  maximum  or  minimum  to  the  one  next  following.  These 
are  known  as  elements  of  variation  and  to  them  is  added  the 


INTRODUCTORY  5 

epoch,  that  is,  the  date  of  some  very  well-determined  maxi- 
mum or  minimum,  its  selection  depending  upon  the  nature 
of  the  curve.  This  date  is  usually  changed  from  the  calendar 
date  into  the  corresponding  day  of  the  Julian  period,  and  is 
known  as  Julian  Day.  The  Julian  period,  as  the  name  implies, 
is  a  continuation  of  that  introduced  at  the  time  of  Julius  Csesar, 
according  to  which  the  days  are  numbered  consecutively,  be- 
ginning at  4713  B.C.  It  has  been  adopted  into  variable  star 
work,  in  order  to  facilitate  the  combination  of  observations 
scattered  over  a  long  period  of  time.  The  Julian  Day  for  Janu- 
ary 1,  1915,  is  2,420,499.  For  a  fuller  explanation  see  Chapter 
XI. 

The  observations  as  described  above  are  finally  combined 
into  one  curve  called  the  mean  light  curve,  which  represents  the 
average  course  of  variation  of  the  star,  smoothing  out  the  small 
irregularities.  It  is  upon  the  study  of  the  mean  light  curves  of 
great  numbers  of  variables  that  the  classification  is  based. 
The  method  of  forming  it  will  be  treated  in  Chapter  X. 

CLASSES  OF  VARIABLES 

From  the  study  of  their  curves,  it  has  been  found  that  vari- 
able stars  may  be  divided  into  distinct  groups,  each  one  hav- 
ing its  own  particular  light  curve.  Several  different  groupings 
have  been  made  by  different  astronomers.  The  one  which  is 
best  known  and  most  widely  used  is  due  to  Professor  E.  C. 
Pickering,  of  the  Harvard  College  Observatory.  It  was  first 
proposed  by  him  in  1880  in  the  Proceedings  of  the  American 
Academy  of  Arts  and  Sciences,  vol.  xvi,  pp.  17,  257.  It  is 
later  repeated  in  the  Provisional  Catalogue  of  Variable  Stars, 
which  forms  No.  in  of  vol.  48  of  the  Annals  of  the  Harvard 
College  Observatory,  from  which  source  the  present  statement 
is  taken. 

Class  I  represents  new  or  temporary  stars;  Class  II,  vari- 
ables of  long  period;  Class  III,  variables  of  small  range,  or 
irregular  variation  according  to  laws  as  yet  unknown;  Class 
IV,  variables  of  short  period;  and  Class  V,  variables  of  the 


6  THE  STUDY  OF  VARIABLE  STARS 

Algol  type.  Class  II  may  be  subdivided  into  Class  Ha,  which 
contains  the  ordinary  variables  of  long  period,  and  Class  lib, 
to  which  U  Geminorum  and  SS  Cygni  belong.  The  latter  are 
usually  faint  and  of  nearly  uniform  brightness,  with  occasional 
sudden  and  irregular  outbursts  of  light  which  diminish  gradu- 
ally. Class  IV  can  similarly  be  divided  into  Class  IVa,  which 
contains  ordinary  variable  stars  of  short  period,  and  Class  IVb, 
of  which  ft  Lyrae  is  the  typical  star. 

CLASS  I.  New  or  Temporary  Stars.  A  new  star  is  one  which 
grows  bright  very  suddenly,  often  in  a  few  hours,  and  then 
fades  away,  more  or  less  gradually,  becoming  either  a  faint 
star,  or  a  planetary  nebula.  An  excellent  discussion  of  these 
stars  may  be  found  in  Miss  Clerke's  interesting  and  valuable 
volumes,  The  System  of  the  Stars  and  Problems  in  Astrophysics. 
The  two  brightest  novae  of  recent  years  were  discovered  by 
Thomas  Anderson  of  Edinburgh.  They  are  Nova  Aurigae  and 
Nova  Persei.  A  portion  of  the  curve  of  the  latter  star  is  given 
below.  Though  it  appears  for  a  time  to  have  somewhat  regu- 
lar fluctuations,  it  is  in  reality  a  variable  having  but  one  maxi- 
mum, which  is  followed  by  a  prolonged  minimum. 

One  might  inquire  whether  it  is  possible  to  obtain  any  in- 
formation regarding  the  history  of  a  new  star  before  the  time 
of  the  first  observation,  and  also  whether  any  such  stars  have 
been  observed  before  their  maximum  brightness  was  attained. 
The  answer  to  both  questions  is  an  affirmative  one.  By  means 
of  the  great  store  of  photographic  plates  at  the  Harvard  Ob- 
servatory and  elsewhere,  it  is  always  possible  to  trace  back  the 
history  of  each  new  star,  until  we  reach  a  time  when  it  is  fainter 
than  any  star  recorded  on  the  photograph.  Just  what  magni- 
tude is  thus  represented  depends  upon  the  length  of  the  photo- 
graphic exposure,  being  sometimes  the  eleventh  magnitude, 
and  sometimes  even  fainter. 

For  example,  Nova  Aurigae,  when  discovered  by  Anderson, 
was  a  yellowish  star  of  the  fifth  magnitude.  From  October  21 
to  December  1, 1891,  photographs  of  the  same  region  had  been 
taken  at  the  Harvard  Observatory,  thirteen  in  number,  from 


8 


THE  STUDY  OF  VARIABLE  STARS 


all  of  which  it  was  absent.  On  December  8  it  is  also  lacking  on 
a  photograph  taken  by  Wolf,  of  Heidelberg,  which  shows  stars 
of  the  ninth  magnitude.  On  December  10,  a  plate  taken  at 
Harvard  shows  it  to  be  of  the  5.4  magnitude,  and  following 
photographs  at  the  same  Observatory  show  that  it  reached  a 
maximum  of  4.4  on  December  20,  hence  it  was  already  on  the 
downward  slope  of  its  light  curve  when  discovered.  Its  sudden 
increase  in  brightness  from  below  the  ninth  magnitude  to  5.4 
must  have  taken  place  in  about  twenty-four  hours.  The  case 
of  Nova  Persei  is  equally  striking.  When  discovered  on  Feb- 
ruary 22,  1901,  it  was  brighter  than  the  second  magnitude  and 
had  not  then  attained  its  greatest  brightness.  On  a  plate  taken 
twenty-eight  hours  previously,  containing  stars  of  the  twelfth 
magnitude,  it  did  not  appear,  hence  it  must  have  increased  ten 
magnitudes  during  that  time. 

A  very  extensive  series  of  observations  covering  the  recent 
history  of  some  of  these  stars  has  been  made  by  Professor 
Barnard  at  the  Yerkes  Observatory,  and  published  in  the 


M 


s 


8.0 
9.0 

100 
1 1.0 


Days      100       £00       300        400        500 


Figure  3 

MEAN  LIGHT  CURVE  OF  T  CASSIOPEIAE 


INTRODUCTORY 


9 


Astronomische  Nachrichten,  No.  4655.  They  include  eleven 
stars  and  cover  a  period  of  twenty  years.  Briefly  stated  the 
result  is  that  some  of  the  novae  are  now  merely  ordinary  faint 
stars,  while  others,  from  their  hazy,  ill-defined  appearance, 
are  regarded  as  probably  nebulous  stars.  Some  are  no  longer 
visible. 

CLASS  II.  Variable  Stars  of  Long  Period.  As  the  name  im- 
plies, the  variation  of  these  stars  is  periodic  in  character,  that 
is,  it  is  repeated  in  prac- 
tically the  same  kind  of 
curve  at  intervals  more 
or  less  regular;  or  in 
other  words,  the  stars 
increase  to  a  maximum 
brightness,  diminish  to  a 
minimum,  and  repeat 
the  process  in  the  same 
manner  periodically. 
There  are  certain  irreg- 
ularities in  the  repeti- 
tions, for  the  brightness 
at  maximum  or  mini- 
mum is  not  uniform, 
neither  is  the  interval  of 
time  between  two  suc- 
cessive maxima  always 
the  same;  nevertheless, 
the  variation  is  distinct- 
ly periodic.  The  greater 
number  of  variables  be- 
long to  this  class.  The 
periods  range  in  length 

from  about  fifty  days  to  over  six  hundred  days,  the  greatest 
number  being  from  two  hundred  and  fifty  to  three  hundred 
days  in  length.  Figures  3  and  4  show  the  mean  light  curves  of 
two  long  period  variables.  They  were  taken  from  the  Harvard 
Observatory  Annals,  vol.  37,  Plate  II. 


D 


<xus 


2,00     300 


Figure  4 


MEAN  LIGHT  CURVE  OF  R  URSAE 
MAJORIS 


10 


THE  STUDY  OF  VARIABLE  STARS 


CLASS  lib.  Stars  of  the  U  Geminorum  Type.  These  are  char- 
acterized by  a  very  rapid  rise  from  a  constant  minimum  to  a 
maximum  which  does  not  last  for  any  regular  interval  of  time, 
being  sometimes  long  and  sometimes  short,  and  which  in  turn 
is  followed  by  a  slow  decline  to  the  minimum.  This  change 
does  not  occur  with  any  regularity,  but  is  always  unexpected. 
There  are  two  other  stars  in  this  class  in  addition  to  the  typical 
one.  They  are  SS  Cygni  and  SS  Aurigae.  The  first  of  these, 
being  a  little  brighter  and  having  a  shorter  period  than  U 
Geminorum,  as  well  as  being  more  favorably  situated  for 
observation,  has  been  studied  more  extensively.  Below  is  given 
its  mean  light  curve,  showing  the  two  forms  of  maximum,  de- 
rived by  Parkhurst  (Astrophysical  Journal,  12,  265). 


Me* 
80 

90 
10.0 
11.0 
12,0 

D 

•\ 

r\ 

\ 

\ 

\ 

\ 

j 

\ 

^ 

ays   2,0      30        40       SO        GO       70        80       90       100 

Figure  5 

LIGHT  CHANGES  OF  SS  CYGNI 

CLASS  III.  Irregular  Variables.  In  this  class  are  placed 
those  stars  which  give  evidence  of  such  irregularities  that  no 
period  can  be  assigned  to  them,  for  example,  a  Orionis  and  a 
Herculis.  It  has  occasionally  happened,  however,  that  a  star 
that  has  been  placed  in  this  class  has  later,  through  more  con- 
tinuous observation,  been  found  to  belong  to  another  group. 
An  example  of  this  is  the  star  u  Herculis,  the  variability  of 
which  was  discovered  as  early  as  1869,  and  which  was  long 


INTRODUCTORY 


11 


classified  with  the  irregular  variables.  In  1908,  from  close 
observation  it  was  found  to  have  a  period  of  2.05  days  and  was 
placed  in  Qass  IV.  Many  of  the  truly  irregular  stars  have  a 
reddish  color,  a  fact  which  is  closely  connected  with  their 
spectra,  and  will  be  explained  more  fully  in  the  section  on 
stellar  spectra. 

CLASS  IV.  Short  Period  Variables.  The  stars  of  this  group 
have  periods  extending  from  several  hours  to  forty-five  days, 
but,  as  will  be  seen  later,  the  actual  dividing  line  between  long 
and  short  period  variables  does  not  depend  on  the  length  of 
period  alone  but  upon  the  spectrum  and  the  character  of  the 
light  curve.  According  to  Pickering's  classification,  this  group 
has  two  subdivisions.  Of  the  first,  8  Cephei  is  the  typical 
star.  It  shows  a  rapid  rise  to  maximum,  which  occupies  in 
general  one-third  of  the  period.  This  is  followed  by  a  slower 
decline,  in  the  course  of  which  there  may  occur  a  more  or  less 
accentuated  halt.  From  the  typical  star  of  the  group  the  name 
"Cepheid"  variables  has  been  in  frequent  use,  but  quite  re- 
cently it  has  been  criticized  on  the  ground  that  it  is  given  also 


Ms 

2..90 
1.10 

> 

-v 

/ 

\ 

/ 

\ 

uo 
2.5-0 
Z4Q 

no 
mo 

MO 

/ 

^ 

\ 

/ 

\ 

' 

/ 

\ 

s 

4 

^S 

/ 

X 

*•% 

\ 

^^ 

/ 

^x 

/ 

^^^ 

•^ 

""^•w. 

S 

Days       1                I               3                4     v         5T               6               7 

Figure  6 

MEAN  LIGHT  CURVE  OF  8  CEPHEI 


12  THE  STUDY  OF  VARIABLE  STARS 

to  meteoric  showers  which  have  their  radiant  points  in  the 
constellation  of  Cepheus.  Hartwig,1  who  makes  the  criticism 
in  his  Katalog  for  1914,  suggests  instead  the  word  "Blink- 
stern,"  for  which  there  is  no  good  English  equivalent.  He  calls 
attention  to  the  fact  that  the  light  curve  of  these  stars  is  char- 
acterized by  a  rapid  brightening  to  a  maximum  of  short  dura- 
tion, which  is  followed  by  a  gradual  decline  to  the  minimum, 
and  furthermore,  that  these  phases  occur  with  extraordinary 
regularity.  He  likens  it  to  the  effect  produced  by  a  revolving 
light  in  a  lighthouse  on  the  sea  coast,  in  which  there  is  the 
same  rapid  brightening,  leading  to  a  short  and  brilliant  maxi- 
mum followed  by  a  gradual  diminution  to  the  minimum, 
always  performed  with  absolute  regularity.  Such  a  light  is 
called  in  German  "Blinklicht,"  whence  the  name  "Blink- 
stern."  However,  it  should  be  said  in  defense  of  the  name 


Mg- 

93 


9.7 


99 


10.1 


10-3 


i  as 


107 


\ 


OHrsI        2.        3        4        S        G        7        8        9       1011 


Figure  7 

MEAN  LIGHT  CURVE  OF  8  ARAB 

1  Ernst  Hartwig,  Katalog  und  Ephemeriden  ver&nderlicher  Sterne  fur  191b 
287. 


INTRODUCTORY 


13 


"Cepheid  Variable"  that  the  term  has  been  in  use  for  some 
years,  and  there  has  been  no  confusion  between  the  variables 
and  the  shooting  stars.  The  group  is  named  from  the  best 
known  star  in  it,  on  exactly  the  principle  according  to  which 
we  refer  to  the  Algol  stars  and  the  ft  Lyrae  type.  The  writer 
sees  no  need  for  the  change. 

On  pages  11  and  12  appear  the  light  curves  of  two  stars  in 
this  group,  differing  somewhat  from  each  other,  the  second 
of  which  has  usually  been  classified  with  a  small  group  called 
Ant-algol  stars  because  their  curves  seem  to  be  an  inversion 
of  the  Algol  type,  but  which  have  been  included  by  Hartwig 
under  the  title  "  Blinkstern."  They  are  8  Cephei 1  and  S  Arae.2 

@  Lyrae  is  typical  of  the  second  division,  and  until  rather 
recently  was  considered  to  be  the  only  representative  of  its 
class,  but  in  1914  the  list  published  in  the  Vierteljahrsschrift 
contained  eighteen  similar  to  it.  The  curve  shows  a  rapid  rise 


Figure  8 

MEAN  LIGHT  CURVE  OF  |3  LYRAE 
1  Joel  Stebbins,  Ap.  J.,  27, 193.      «  Alex.  W.  Roberts,  Ap.  J.,  33,  208. 


14 


THE  STUDY  OF  VARIABLE  STARS 


from  minimum  to  maximum,  similar  to  that  of  S  Cephei,  fol- 
lowed by  a  decline  to  a  secondary  minimum,  after  which  there 
is  another  rise  to  a  maximum  of  the  same  magnitude  as  before, 
and  again  a  decline  to  the  primary  minimum.  That  is  to  say, 
there  are  two  equal  maxima  separated  by  two  unequal  minima. 
The  star's  curve  is  given  on  page  13. l 

There  are  a  few  short  period  variables  which  show  a  sym- 
metrical curve,  that  is,  one  whose  ascending  and  descending 
branches  are  alike. 

£  Geminorum  is  the  typical  star  of  the  group.  Its  light 
curve2  is  given  below.  Hartwig  mentions  nine  which  he  calls 
of  this  type. 


Ms 

3.7 
4-5 

Da 

x 

^ 

** 

^ 

k 

< 

a*lZ34£G789lO 

Figure  9 

LIGHT  CURVE  OF  <  GEMINORUM 

What  is  known  as  the  cluster  type  of  variable  will  be  dis- 
cussed in  Chapter  XII.  Typical  curves  may  be  found  in  An- 
nals,  H.C.O.,  vol.  38.  The  one  which  occurs  with  greatest 
frequency  is  similar  to  the  curve  of  S  Arae  shown  above. 

CLASS  V.  Algol  Stars.  The  stars  of  this  group  also  have 
short  periods  and  hence  might  properly  form  a  subdivision  of 
Class  IV,  but  they  are  so  numerous  and  have  such  a  distinc- 
tive curve  that  they  are  grouped  together,  and  take  their  name 
from  the  first  representative  discovered.  The  curve  is  charac- 

i  G.  W.  Myers,  Ap.  J.,  7,  3.  a  W.  W.  Campbell,  Ap.  J.,  13,  92. 


INTRODUCTORY 


15 


terized  by  a  sustained  maximum  broken  by  a  swift  descent  to 
minimum,  which  is  sometimes  quite  short,  and  sometimes  con- 
tinues for  an  hour  or  so.  Following  is  a  rise  to  maximum  very 
nearly  symmetrical  with  the  descending  branch.  The  time 
spent  in  the  change  is  called  the  duration  of  phase.  In  some 
cases  a  secondary  minimum  appears.  Below  are  given  the 
light  curves  of  two  stars  in  this  group;  that  of  Algol,1  which  has 
a  short  minimum  and  that  of  U  Cephei,2  which  has  a  long 
minimum. 
The  descriptions  just  given  are  necessarily  brief  and  lacking 


CKange 


M 


Hours 


GO 


Figure  10 

LIGHT  CURVE  OF  ALGOL 


1  Joel  Stebbins.  Ap.  J.,  32,  199.        2  P.  S.  Yendell,  Pop.  Ast.,  14,  600. 


MJ 


Figure  11 

MEAN  LIGHT  CURVE  OF  U  CEPHEI 


INTRODUCTORY  17 

in  details,  but  are  sufficient  for  introductory  purposes.  They 
should  enable  the  reader  to  understand  the  characteristics  of 
the  different  classes  of  variables  and  to  recognize  their  light 
curves.  Many  variables  are  as  yet  unclassified  because  suffi- 
cient data  have  not  been  gathered  to  enable  us  to  determine 
the  light  curves.  These  are  designated  as  unknown,  or  in  Ger- 
man, "unbekannt."  It  should  finally  be  stated  that  recently 
there  has  been  a  tendency  to  give  up  the  distinctions  between 
some  of  the  different  classes  of  short  period  variables,  and  to 
put  the  Algol  and  /3  Lyrae  variables  under  one  heading,  since 
their  variation  is  supposed  to  be  due  to  the  same  cause.  Cer- 
tain individual  stars  also,  upon  closer  examination  of  their 
curves,  have  been  transferred  from  one  division  to  another. 
The  present  classification  cannot  be  considered  as  final. 

PRINCIPLES   OF  SPECTRUM  ANALYSIS 

It  has  been  found  from  a  statistical  study  of  the  spectra  of 
variable  stars  that  there  is  a  marked  correlation  between  the 
character  of  the  spectrum  and  the  type  of  variation,  and  also 
that  with  certain  types  of  stars  there  are  important  changes 
that  take  place  in  the  spectrum  accompanying  the  changes  in 
the  light.  In  order  to  interpret  these  correctly,  it  will  be  neces- 
sary to  understand  the  principles  which  underlie  the  formation 
of  the  spectrum  and  to  be  familiar  with  the  classification  of 
stellar  spectra. 

If  we  allow  a  beam  of  white  light  such  as  comes  from  a  can- 
dle or  an  incandescent  light  to  fall  upon  a  prism,  we  find  that 
when  it  emerges  from  the  prism,  instead  of  being  white  it  is 
broken  up  into  a  rainbow  band  of  colors  arranged  in  the  order: 
red,  orange,  yellow,  green,  blue,  indigo,  and  violet.  We  find 
also  that  the  ray  of  light  has  changed  its  direction,  that  it  is 
bent  aside  from  the  path  it  would  pursue  if  the  prism  were  re- 
moved, that  the  red  color  is  bent  the  least,  and  the  violet  the 
most.  The  breaking  up  of  the  light  into  its  component  colors 
is  called  dispersion,  the  bending  of  the  different  colors,  refrac- 
tion, and  the  angle  through  which  it  takes  place  is  called  the 


18 


THE  STUDY  OF  VARIABLE  STARS 


angle  of  deviation.  The  purest  colors  in  the  spectrum  are  ob- 
tained when  the  light  which  falls  upon  the  prism  is  admitted 
through  a  narrow  slit,  and  for  the  best  effect  the  slit  should  be 
placed  a  long  distance  away  from  the  prism.  There  are  obvi- 
ous practical  difficulties  in  the  way  of  doing  this,  and  the  same 
result  is  accomplished  by  placing  the  slit  in  the  focus  of  a  tele- 
scope which  is  turned  toward  the  prism.  The  rays  of  light 
then  emerge  parallel  from  the  lens  and  according  to  the  well- 
known  laws  of  optics  the  effect  is  produced  of  removing  the 
slit  to  a  great  distance.  This  is  called  the  collimating  tele- 
scope. In  order  to  examine  the  spectrum  to  the  best  advantage, 
another  telescope  is  added  called  the  view  telescope,  which  is 
equipped  with  a  micrometer  and  cross  wires  for  making  meas- 
urements. 

The  accompanying  diagram  shows  a  simple  laboratory  prism 
spectroscope.    To  the  parts  mentioned  above  is  added  a  scale 


Figure  12 

PRISM  SPECTROSCOPE 

for  reading  the  angle  of  deviation  of  any  desired  part  of  the 
spectrum.  When  the  spectrum  of  a  star  is  to  be  observed,  the 
spectroscope  must  be  attached  to  the  telescope,  and  should 
be  especially  designed  for  the  purpose  to  which  it  is  to  be  put. 


INTRODUCTORY  19 

In  order  to  understand  what  causes  light  to  behave  in  this 
way  and  how  the  spectrum  may  be  used  for  measurement,  it 
will  be  necessary  to  give  some  explanation  of  the  principles  of 
light.  It  should  be  understood,  however,  at  the  outset,  that 
the  explanation  is  not  exhaustive,  but  is  intended  only  to  give 
a  sketch  of  the  problem,  and  those  who  are  interested  to  pur- 
sue it  further  are  referred  to  the  various  standard  authorities 
on  the  subject. 

We  first  assume  the  existence  of  ether,  a  transparent  elastic 
medium,  filling  all  space  and  even  the  interstices  of  matter. 
Light  is  a  wave  motion  in  this  ether,  and  is  transmitted  through 
it  in  straight  lines.  This  is  called  the  rectilinear  propagation 
of  light.  The  wave  motion  or  vibration  is  a  periodic  disturb- 
ance which  is  handed  on  successively  from  one  portion  of  the 
medium  to  another.  The  particle  which  is  being  disturbed  has 
a  vibratory  motion  but  does  not  travel  on  with  the  wave.  The 
simplest  illustration  of  this  is  found  in  water  waves.  If  a  stone 
is  thrown  into  a  quiet  pool  of  water,  a  series  of  waves  is  started 
which  spread  in  ever  widening  circles  to  the  edges  of  the  pool. 
Those  nearest  the  center  of  disturbance  are  the  most  violent, 
and  the  depth  of  the  successive  waves  diminishes  as  their  dis- 
tance from  it  increases.  The  particles  of  water  do  not  travel 
outward  with  the  waves,  for  if  a  leaf  is  floating  on  the  pool  it 
will  not  be  driven  to  the  shore,  but  will  rise  and  fall  on  the 
surface  of  the  waves  as  they  pass  under  it.  Its  direction  of 
vibration  is  thus  perpendicular  to  the  direction  of  propagation 
of  the  waves. 

Ether  waves  are  of  the  same  nature,  that  is,  the  vibration  of 
their  particles  is  perpendicular  to  the  direction  of  propagation. 
Furthermore,  so  perfect  is  the  elasticity  of  the  ether  that  any 
number  of  waves  can  pass  through  it  at  the  same  time  in  all 
directions,  without  interfering  with  one  another. 

A  wave-length  is  the  distance  the  disturbance  travels  while 
the  first  particle  is  executing  one  vibration.  This  is  generally 
represented  by  the  symbol  X. 

The  accompanying  diagram  shows  the  different  parts  of  a 


20  THE  STUDY  OF  VARIABLE  STARS 

complete  wave.  The  arrows  attached  to  the  dots  represent 
the  direction  in  which  each  particle  is  about  to  move.  In  the 
diagram,  a,  the  first  particle,  has  completed  one  vibration,  b 
three  fourths  of  a  vibration,  c  one  half,  and  d  one  fourth,  while 
e  is  about  to  begin  its  oscillation.  The  distances  of  b  and  d  from 
the  line  represent  the  maximum  displacement  in  the  wave, 
and  are  called  the  amplitude  of  the  vibration.  The  distance 
ae  is  called  the  wave-length.  The  phase  of  a  particle  is  defined 
by  its  position  in  the  wave,  that  is,  by  its  distance  from  the 
original  position  and  its  direction  of  motion.  This  may  be 


Figure  13 

DIAGRAM  OF  A  COMPLETE  WAVE 

more  clearly  understood  by  pointing  out  when  two  particles 
are  in  the  same  or  in  opposite  phase.  The  former  occurs  when 
the  two  particles  have  the  same  displacement  and  the  same  di- 
rection of  motion;  for  example,  a  and  e,  for  a  has  completed  one 
vibration,  and  is  ready  to  start  on  a  second,  while  e  is  about 
to  begin  its  first  vibration.  They  are  separated  by  a  whole 
wave-length.  On  the  other  hand,  6  and  d  are  in  opposite  phase, 
for  though  they  have  the  same  displacement  in  amount,  they 
have  opposite  directions  of  motion,  since  b  is  about  to  move  up 
toward  its  original  position,  while  d  is  about  to  move  down. 
They  are  separated  by  an  odd  number  of  half  wave-lengths. 

The  length  of  the  light  wave  which  falls  upon  the  retina 
determines  the  sense  impression  of  color,  and  different  por- 
tions of  the  spectrum  have  different  wave-lengths.  The  actual 
length  of  a  wave  of  any  isolated  color  may  be  obtained  in  the 
physical  laboratory  with  the  use  of  a  spectrometer.  The  length 


INTRODUCTORY  21 

of  the  wave  of  a  red  ray  at  the  extreme  end  of  the  spectrum 
is  0.0007600  millimeters,  and  that  of  the  extreme  violet 
0.0003900  millimeters.  Since  these  numbers  are  inconvenient 
to  handle,  a  small  unit  has  been  adopted  called  the  Angstrom 
unit  from  the  name  of  the  Swedish  physicist  who  first  used  it. 

It  has  a  value  of  -—  meter  or  a  tentnmeter,  and  is  abbreviated 

A.U.  Measurements  of  wave-lengths  can  be  made  with  ex- 
treme accuracy,  that  of  the  sodium  line  D  being  5896.616  A.U. 

There  are  other  portions  of  the  spectrum  beyond  the  region 
visible  to  the  eye.  That  part  having  a  shorter  wave-length 
than  3900  A.U.  is  called  the  ultra-violet,  and  can  be  exten- 
sively studied  by  photography,  as  the  rays  possess  strong 
actinic  power.  Many  stars  are  strong  in  this  part  of  the  spec- 
trum. The  infra-red  lies  at  the  other  end  of  the  visible  spec- 
trum, beyond  wave-length  7600.  It  has  strong  heat  radiation 
and  its  character  may  be  studied  by  means  of  its  thermal 
effects.  The  infra-red  spectrum  of  the  sun  has  been  thoroughly 
investigated  since  its  heat  emanations  are  very  strong,  while 
this  part  of  stellar  spectra  is  lacking,  since  only  a  minute  quan- 
tity of  heat  reaches  the  earth  from  the  stars. 

The  vibration  frequency  of  an  oscillating  particle,  or  the 
number  of  vibrations  which  strike  the  eye  per  second,  is  of 
importance  and  may  be  found  as  follows :  — 

Let  X  =  the  wave-length  or  the  distance  the  disturbance 
has  traveled  while  the  original  particle  has  exe- 
cuted one  vibration; 
V  =  the  distance  traveled  by  light  during  one  second 

of  time,  or  186,330  miles; 

n  =  the  number  of  vibrations  performed  by  a  particle 
during  one  second; 

Then  n  =  Xr. 

A/ 

The  vibration  frequency  thus  bears  an  inverse  relation  to 
the  wave-length.  In  the  red  end  of  the  spectrum, 

n  =  ^^  mileS  =  395  000  000  000  000. 
.OOUToU  mm 


22  THE  STUDY  OF  VARIABLE  STARS 

For  the  violet  end  of  the  spectrum,  it  is  760  000  000  000  000. 

As  has  been  stated,  when  white  light  passes  through  a  prism, 
it  appears  as  a  band  of  color  in  which  there  is  no  break  in  the 
series  from  the  red  at  one  end  to  the  violet  at  the  other.  Such 
light  comes  from  a  candle  flame  or  from  an  incandescent  light, 
and  the  spectrum  that  it  produces  is  called  a  continuous  spec- 
trum. If  the  source  of  light  should  happen  to  be  a  Bunsen 
flame  in  which  sodium  chloride  or  common  salt  is  burning,  the 
appearance  presented  by  the  spectrum  would  be  quite  differ- 
ent. Instead  of  showing  the  unbroken  band  of  color,  there 
would  be  seen  two  yellow  lines  very  close  together  and  noth- 
ing else.  Further,  when  we  examine  the  spectrum  of  the  sun, 
we  see  the  rainbow  band  of  color,  but  it  is  crossed  by  vertical 
dark  lines  arranged  in  irregular  groups,  which  are  invariably 
the  same  at  all  times  of  observation.  It  appears,  thus,  that 
the  character  of  the  spectrum  depends  upon  the  nature  and 
condition  of  the  substance  or  body  producing  it.  From  the 
investigation  of  these  facts,  the  principles  of  spectrum  analy- 
sis have  been  deduced.  Their  importance  cannot  be  overesti- 
mated, since  upon  a  correct  understanding  of  them  depends  the 
interpretation  of  the  many  details  presented  by  the  spectra 
of  the  sun  and  stars.  They  may  be  expressed  in  the  following 
simple  forms,  which  must  be  understood  as  being  merely  ab- 
breviated statements  and  not  an  explanation  of  the  complete 
laws. 

A  bright  spectrum,  whether  it  be  continuous,  or  whether 
it  consist  of  separate  bright  lines,  is  called  an  emission  spec- 
trum. When  the  bright  spectrum  is  crossed  by  dark  lines  or 
bands,  the  lines  and  bands  together  form  an  absorption  spec- 
trum. An  emission  spectrum,  then,  may  be  seen  by  itself,  but 
an  absorption  spectrum  is  only  seen  when  superposed  upon  a 
bright  background,  that  is,  against  an  emission  spectrum.  A 
further  investigation  of  the  lines  shows  the  following  general 
principles:  — 

1.  An  incandescent  solid  or  liquid  or  a  gas  under  very  high 
pressure  will  give  a  continuous  spectrum. 


INTRODUCTORY  23 

2.  An  incandescent  gas  will  give  a  discontinuous  spectrum, 
that  is,  a  spectrum  consisting  of  separate  bright  lines.   The 
lines  forming  this  spectrum  invariably  occupy  the  same  posi- 
tions, that  is,  have  the  same  wave-lengths,  so  long  as  the  con- 
ditions of  temperature  and  pressure  affecting  the  source  of 
light  remain  the  same.    For  example,  under  ordinary  condi- 
tions the  spectrum  of  hydrogen  consists  of  four  bright  lines, 
one  red,  one  bluish  green,  and  two  violet,  called  in  order  Ha, 
H/3,  HY,  HS.  Each  gas,  then,  has  its  own  particular  spectrum, 
and  the  wave-lengths  of  its  lines  can  be  measured.   No  two 
gases  have  lines  of  the  same  wave-length;  that  is,  there  is  no 
line  common  to  two  gases.   The  spectra  of  almost  all  of  the 
known  elements  on  the  earth's  surface  have  been  investigated 
and  mapped,  and  tables  of  wave-lengths  have  been  published. 
It  is  no  easy  task  to  complete  this  investigation,  partly  because 
some  of  the  elements  are  quite  rare,  but  primarily  because  the 
spectrum  varies  with  changing  temperature  and  pressure,  as 
will  be  described  at  length  later  on.  It  is  obvious  that  the  ele- 
ments represented  in  a  continuous  spectrum  cannot  be  distin- 
guished. 

3.  A  gas  absorbs  from  white  light  passing  through  it  pre- 
cisely those  wave-lengths  of  which  its  own  spectrum  consists. 
If  it  is  a  cool  gas,  or  a  luminous  one  but  of  a  lower  temperature 
than  the  source  of  white  light  behind  it,  it  will  produce  rela- 
tively dark  lines  in  the  spectrum.  If  it  is  hotter  than  the  source 
behind  it,  it  will  produce  bright  lines.  If  of  just  the  same  tem- 
perature, no  effect  will  be  produced.    This  third  principle  is 
known  as  Kirchhoff's  law. 

Thus  we  can  determine  to  some  extent  from  the  appearance 
of  the  spectrum  of  a  heavenly  body  what  its  physical  condition 
is,  and  of  what  elements  it  is  composed.  If  like  the  sun  it  has 
a  continuous  spectrum  crossed  by  dark  lines,  we  know  that  it 
consists  of  a  central  core  which  produces  a  continuous  spec- 
trum; that  is,  it  must  be  a  glowing  solid  or  liquid,  or  a  gas 
under  very  great  pressure,  but  we  cannot  tell  which  of  the 
three,  nor  can  we  tell  of  what  elements  it  is  composed.  We 


24  THE  STUDY  OF  VARIABLE  STARS 

know  further,  that  this  central  core  is  surrounded  by  an  at- 
mosphere of  cooler  gases  which  we  can  identify  after  the  wave- 
lengths have  been  measured,  by  their  coincidence  with  the 
wave-lengths  of  terrestrial  substances,  if  these  are  already 
known. 

If,  on  the  other  hand,  as  is  the  case  with  some  of  the  stars, 
the  continuous  spectrum  is  crossed  by  bright  lines,  this  is  an 
indication  that  the  central  core  is  surrounded  by  an  atmos- 
phere which  is  of  a  higher  temperature  than  itself;  and  as  be- 
fore, if  we  can  identify  the  lines,  we  can  identify  the  elements 
of  which  the  atmosphere  is  composed. 

A  discontinuous  spectrum  of  bright  lines,  which  is  character- 
istic of  certain  of  the  nebulae,  shows  that  the  body  producing 
it  is  a  true  gas. 

The  above  statements  make  it  evident  that  before  we  can 
decide  what  elements  are  present  in  the  atmosphere  of  the  sun 
or  a  star,  we  must  have  a  complete  set  of  the  wave-lengths 
peculiar  to  all  of  the  elements  on  the  earth's  surface. 

From  the  second  principle  of  spectrum  analysis  it  follows 
that  in  order  to  present  a  spectrum  of  separate  bright  lines,  the 
substance  must  be  in  the  form  of  an  incandescent  gas.  Some 
elements,  such  as  hydrogen  and  oxygen,  exist  as  permanent 
gases,  but  others  must  be  subjected  to  heat  in  order  to  become 
volatilized.  The  temperature  at  which  this  occurs  varies  with 
the  different  elements,  and  hence  quite  different  methods  must 
be  employed  in  order  to  render  them  incandescent. 

Furthermore,  the  second  part  of  this  principle  states  that  the 
lines  in  the  spectrum  are  the  same  under  the  same  conditions 
of  temperature  and  pressure,  implying  that  when  the  conditions 
are  changed,  the  appearance  of  the  spectrum  changes  also. 

In  view  of  these  facts,  it  becomes  necessary  to  make  the 
investigation  very  exhaustive  and  to  vary  the  conditions  of 
temperature  and  pressure  in  every  possible  way.  Researches 
of  this  kind  have  been  carried  on  during  many  years  and  the 
wave-lengths  of  lines  in  the  emission  spectra  of  the  elements 
are  extensively  but  not  completely  known,  since  not  all  of  the 


INTRODUCTORY  25 

possible  variations  of  temperature  and  pressure  have  been 
applied  to  them.  Neither  have  all  of  the  effects  which  we  see 
in  stellar  spectra  been  obtained,  because  we  are  unable  to  re- 
produce even  approximately  the  conditions  which  exist  in  the 
stars  with  their  enormous  masses  and  small  densities.  The  re- 
sults which  have  been  obtained  are  so  varied  and  in  some  cases, 
so  unexpected,  that  it  is  not  yet  possible  to  state  general  rela- 
tions in  a  definite  form.  At  the  same  time  certain  facts  are 
well  substantiated  and  those  which  bear  directly  on  our  sub- 
ject should  be  noted. 

The  several  methods  of  rendering  a  solid  substance  lumi- 
nous may  be  divided  into  the  following  classes,  the  order 
representing  the  relative  temperature:  the  Bunsen  flame,  the 
oxy-hydrogen  blowpipe  flame,  the  electric  furnace,  the  elec- 
tric arc,  and  the  electric  spark  of  different  intensities.  For 
the  gases,  still  another  method  is  used:  they  are  confined  in 
vacuum  tubes,  and  made  luminous  by  the  passage  of  a  spark 
under  low  pressure. 

The  spectra  produced  by  these  means  are  in  general  of  two 
kinds,  banded  and  lined.  The  former  usually  consists  of  a  num- 
ber of  bands,  each  having  one  bright  edge  and  gradually  di- 
minishing almost  to  darkness  in  the  direction  of  the  other  edge. 
Under  high  dispersion  each  band  is  resolved  into  numerous 
slender  lines,  very  closely  packed  together,  among  them  being 
at  certain  rhythmic  intervals,  brighter  lines,  giving  the  effect 
of  a  fluted  column  brightly  illuminated  so  that  the  grooves  are 
in  shadow.  From  this  appearance  is  derived  the  name  "chan- 
neled" or  "fluted"  which  is  often  applied  to  a  spectrum. 
Band  spectra  are  produced  by  compounds,  such  as  titanium 
oxide,  and  also  by  elements  at  temperatures  below  that  neces- 
sary for  the  production  of  lines.  Perhaps  the  most  beautiful 
banded  spectrum  is  that  due  to  carbon,  which  is  produced 
when  the  simple  arc  light  is  examined  with  a  spectroscope. 
As  this  also  appears  whenever  the  arc  is  used  in  producing  the 
spectrum  of  any  other  substance,  a  knowledge  of  it  is  of  ex- 
treme importance,  and  hence  it  has  been  very  extensively 


26  THE  STUDY  OF  VARIABLE  STARS 

studied.  There  are  several  bands  in  different  parts  of  this 
spectrum,  each  one  of  which  consists  of  several  edges,  while  the 
background  is  filled  in  with  multitudes  of  very  fine  lines 

The  line  spectrum  consists  of  isolated  lines.  The  variations 
in  it  come  from  differences  in  temperature  and  also  from  the 
manner  in  which  it  is  produced.  The  effect  of  a  change  in  tem- 
perature in  passing  from  the  flame  spectrum  of  an  element  to 
that  of  the  arc,  is  to  increase  the  number  of  lines;  but  the 
effect  of  a  spark  discharge,  especially  when  of  great  intensity, 
is  to  change  the  relative  intensity  of  the  lines  in  the  spectrum. 
This  important  fact  was  discovered  by  Lockyer,  and  may  be 
described  as  follows.  Certain  lines  in  the  spectrum  of  an  ele- 
ment as  produced  by  the  electric  arc  are  shorter  than  the 
others,  that  is,  they  do  not  extend  over  the  entire  length  of 
the  slit.  In  passing  from  the  arc  to  the  spark,  these  lines  be- 
come greatly  strengthened,  and  as  the  tension  of  the  spark  is 
increased,  these  lines  become  the  strongest  in  the  spectrum  and 
the  fainter  lines  tend  to  disappear,  leaving  as  a  result  a  simpler 
spectrum,  consisting  only  of  the  brightest  lines.  These  lines 
are  called  by  Lockyer  "enhanced"  lines,  and  are  thought  by 
him  to  be  due  to  a  very  high  temperature.  Hence,  whenever 
they  are  found  in  a  stellar  spectrum,  they  indicate  that  it  is 
produced  by  a  star  of  very  high  temperature.  Other  investi- 
gators question  whether  the  presence  of  these  lines  is  due  to 
an  enormously  hot  temperature,  and  suggest  that  it  may  arise 
from  the  conditions  of  stress  existing  in  the  spark  itself. 

The  pressure  existing  in  the  gas  under  investigation  has  an 
important  effect  on  its  spectrum.  An  increase  of  pressure  will 
widen  the  lines,  so  that  with  a  sufficient  amount  they  will  be- 
come so  wide  as  to  coalesce  and  form  a  continuous  spectrum. 
In  the  spectrum  produced  by  the  arc,  the  lines  often  appear 
arrow-shaped,  showing  that  in  the  lower  part  of  the  carbon  cup 
the  density  of  the  gas  is  greater  than  it  is  farther  up.  It  has 
also  been  found  that  a  very  high  pressure  will  shift  the  lines 
slightly  toward  the  red  end  of  the  spectrum. 

The  presence  of  a  strong  magnetic  field  about  the  incandes- 


INTRODUCTORY  27 

cent  substance  will  cause  the  lines  to  be  widened  and  split 
into  component  parts,  the  number  and  intensity  of  which  de- 
pend upon  the  element,  the  particular  line,  and  the  strength 
of  the  current.  This  is  known  as  the  Zeeman  effect. 

Another  change  in  the  appearance  of  spectral  lines  is  known 
as  reversal.  This  means  the  change  of  a  line  from  dark  to 
bright,  or  vice  versa.  It  occurs  when  a  gas  of  different  tem- 
perature from  that  producing  the  line  is  temporarily  thrown  in 
front  of  it.  More  often  a  portion  of  a  line  will  be  reversed  in- 
stead of  the  entire  line.  It  is  observed  frequently  in  the  spec- 
trum of  the  sun,  particularly  with  certain  lines.  These  are 
ordinarily  dark,  but  under  certain  circumstances,  a  bright 
reversal  will  appear,  indicating  that  there  is  a  sudden  outburst 
of  hotter  gas  projected  in  front  of  the  cooler  gas  which  pro- 
duced the  dark  line. 

A  few  words  may  be  said  about  the  different  methods  of 
producing  the  spectrum.  When  the  Bunsen  flame  is  used,  a 
salt  of  the  element  to  be  studied  is  placed  in  the  gas  flame. 
Sometimes  the  solid  substance  is  used  on  a  platinum  wire, 
and  sometimes  a  solution  is  fed  into  the  flame  a  little  at  a  time. 
With  the  oxy-hydrogen  blowpipe  the  apparatus  is  arranged 
so  that  the  flame  plays  upon  the  metallic  salt,  and  later  passes 
out  of  an  opening  where  it  can  be  examined.  With  the  electric 
arc,  three  methods  can  be  used.  For  the  lower  carbon,  one 
with  a  soft  core  is  taken,  the  core  is  dug  out  and  some  of  the 
metallic  salt  packed  in  its  place.  The  substance  may  also  be 
fed  into  the  arc  in  small  quantities.  This,  however,  is  likely 
to  cause  sudden  changes  and  disturb  the  conditions.  Some- 
times metallic  electrodes  are  used,  if  the  metal  is  not  too  easily 
fusible.  This  method  is  especially  applicable  to  iron.  With 
the  spark,  metallic  terminals  may  be  used,  or  the  spark  may 
be  made  to  pass  from  a  platinum  terminal  to  a  solution  of  the 
salt.  The  work  with  the  electric  furnace,  which  has  been  de- 
veloped comparatively  recently,  does  not  give  quite  so  high  a 
temperature  as  the  arc,  but  it  presents  several  advantages.1 
i  A.  S.  King,  Ap.  J.,  28,  300. 


28  THE  STUDY  OF  VARIABLE  STARS 

A  long  steady  column  of  vapor  can  be  ob- 
tained, the  temperature  can  be  regulated 
without  altering  other  conditions,  the  spec- 
trum gives  almost  as  many  lines  as  the  arc, 
the  control  of  the  conditions  is  greater,  so 
that  changes  in  pressure  can  be  observed, 
and  the  effects  of  absorption  may  be  inves- 
tigated by  using  a  source  of  white  light 
behind  the  vapor  in  the  furnace.  The  labo- 
ratory of  the  Mt.  Wilson  Solar  Observatory 
is  provided  with  an  elaborate  equipment 
for  studying  the  spectra  as  produced  by 
the  electric  furnace,  and  its  connection  with 
the  Observatory  makes  its  work  bear  di- 
rectly upon  the  problems  presented  by  solar 
and  stellar  spectroscopy. 

Another  important  problem  connected 
with  the  study  of  stellar  spectra  has  to 
do  with  the  shifting  of  the  lines  due  to 
motion  in  the  line  of  sight.  This  is  ex- 
plained by  Doppler's  principle,  which  will 
be  treated  fully  in  the  chapter  on  spec- 
troscopic  binaries. 


CLASSIFICATION   OF   STELLAR   SPECTRA 

The  next  step  will  be  to  classify  the 
spectra  of  the  stars,  since  it  has  been  found 
that  certain  types  of  spectra  are  character- 
istic of  certain  types  of  variable  stars. 
Preliminary  to  this,  it  will  be  of  much  as- 
sistance to  give  first  a  description  of  the 
spectrum  of  the  sun,  since  the  lines  and 
groups  which  are  most  prominent  in  it  are 
conspicuous  also  in  the  stars.  Following 
the  description  is  a  table  giving  the  wave- 
lengths of  the  single  lines,  and  of  the 


FlO.  14.   THE  SOLAR 
SPECTRUM 


INTRODUCTORY  29 

strongest  line  in  each  group.     The  relative  intensities  and  the 
sources  of  the  lines  are  also  included. 

A,  a  group  of  heavy  lines  in  the  extreme  red. 
a,  a  group  of  fainter  lines  following  A. 

B,  a  third  group  of  lines  in  the  red. 

C,  a  single  line  in  the  red  toward  the  orange. 

D1?  D2,  a  pair  of  lines  close  together  in  the  yellow. 

E,  a  group  of  fine  lines  in  the  light  green. 

bi,  b2,  bs,  a  group  of  three  sharp  lines  farther  on  in  the 
green. 

F,  a  single  line  in  the  bluish  green. 

Hy,  a  single  line  in  the  blue,  closely  preceding  the  following 
group. 

G,  a  strongly  marked  group  of  lines  in  the  blue, 
g,  a  single  strong  line  in  the  violet. 

h,  a  single  line  also  in  the  violet. 
H,  K,  two  very  heavy  lines  in  the  extreme  violet. 
These  dark  lines  are  known  as  the  Fraunhofer  lines,  for  it 
was  he  who  first  studied  them  and  gave  them  their  names. 


Line 

Wave-Length 

Intensity 

Source 

A 

7594 

Group 

Atmospheric 

a 

7164 

Group 

« 

B 

6870 

14 

Atmospheric  oxygen 

C 

6563 

40 

Hydrogen                      Ha 

Di 

5896,  D2  5890 

20,  30 

Sodium 

E 

5270 

8 

Iron 

bi 

5184,  D25173, 

30,  20, 

Magnesium 

b3  5167 

20 

F 

4861 

30 

Hydrogen                     H/3 

HT 

4341 

20 

Hydrogen                     Hy 

G 

4308 

10 

Iron 

g 

4227 

20 

Calcium 

h 

4102 

40 

Hydrogen                     H5 

H 

3969 

700 

Calcium 

K 

3934 

1000 

Calcium 

30  THE  STUDY  OF  VARIABLE  STARS 

The  earliest  classification  of  stellar  spectra  is  due  to  Father 
Secchi,  an  Italian  astronomer  stationed  at  the  Collegio  Ro- 
mano. It  was  published  in  1849,  but  he  was  not  the  first  to 
perceive  that  differences  existed,  for  Fraunhofer,  in  1823,  had 
already  recognized  the  fact.  Secchi's  classification,  which  was 
made  with  a  small  telescope,  rests  on  a  study  of  the  visual  part 
of  the  spectrum,  and  includes  only  stars  with  absorption  spec- 
tra, i.e.,  those  showing  dark  lines.  It  may  be  briefly  described 
as  follows:  — 

Type  I.  This  is  characterized  by  the  presence  of  four  very 
intense  lines  identified  as  belonging  to  hydrogen.  The  contin- 
uous spectrum  is  rich  in  blue  and  violet  light  and  the  stars 
are  therefore  white  in  color.  Sirius  is  the  most  brilliant  ex- 
ample of  the  type,  for  which  reason  it  is  frequently  called  the 
Sirian  type.  Vega  and  Regulus  also  belong  to  this  type. 

Type  II.  This  has  a  spectrum  resembling  that  of  the  sun, 
consisting  of  many  fine  lines,  with  the  predominating  H  and 
K.  It  is  called  the  solar  type,  examples  among  the  bright  stars 
being  Capella,  Pollux,  and  Arcturus.  Since  the  lines  are  quite 
thickly  massed  in  the  blue  end  of  the  spectrum,  the  resulting 
color  of  the  stars  is  yellowish. 

Type  III.  This  type  is  marked  by  the  presence  of  bands 
which  are  sharply  defined  on  the  side  toward  the  violet  and 
shade  away  toward  the  red.  There  is  strong  general  absorp- 
tion in  the  violet  end,  hence  the  stars  in  this  class  are  pro- 
nouncedly reddish  in  color.  Betelgeuse  is  the  typical  star. 

Type  IV.  These  stars  are  also  characterized  by  bands,  which 
however  shade  toward  the  blue  instead  of  the  red.  They  are 
also  very  red,  but  are  quite  faint,  the  brightest  being  not  more 
than  fifth  magnitude.  A  typical  star  is  152  Schjellerup. 

At  a  later  time  Pickering  suggested  the  addition  of  a  fifth 
type  which  should  include  stars  having  bright  lines  in  their 
spectra.  This  is  known  as  Pickering's  Type  V. 

While  this  classification  was  generally  satisfactory  and  is 
still  used  for  quick  reference,  it  depended  entirely  upon  vis- 
ual observations  made  with  an  instrument  of  small  dispersing 


INTRODUCTORY  31 

power.  When  the  photographic  process  was  employed  for  re- 
cording stellar  spectra,  the  resulting  plates  showed  that  in 
general  Secchi's  classification  held,  but  that  it  was  susceptible 
of  many  fine  gradations  which  could  be  arranged  in  such  a  way 
as  to  show  an  orderly  development  from  one  type  to  another. 
The  principal  fact  upon  which  the  new  classification  was  based 
was  that  certain  groups  of  lines  seemed  to  appear  together 
and  to  act  in  common,  that  is,  to  grow  more  intense  together 
or  to  grow  faint  at  the  same  time.  This  does  not  mean  that  the 
lines  in  a  given  group  all  have  the  same  intensity,  but  that  all 
change  in  the  same  fashion.  The  classification,  which  has 
received  the  general  approval  of  astronomers,  was  developed 
at  the  Harvard  Observatory  and  is  based  upon  the  study  of  a 
great  number  of  spectrograms.  A  detailed  account  of  it  is 
given  in  volume  28  of  the  Annals,  and  was  prepared  for  pub- 
lication by  Miss  Annie  J.  Cannon,  who  has  had  a  larger 
experience  in  dealing  with  stellar  spectra  than  any  other 
astronomer.  Starting  with  the  spectra  of  nebulae,  which  con- 
tain bright  lines  and  are  supposed  to  be  at  the  earliest  stage  of 
development,  and  continuing  with  the  bright  line  stars,  the 
groups,  which  are  denoted  by  letters,  succeed  each  other  in 
order  through  the  white  stars  to  the  yellow  and  red.  It  hap- 
pened that  beginning  with  Secchi's  first  type,  the  separate 
groups  were  lettered  before  they  were  put  in  the  order  of  their 
evolution,  and  the  bright  line  stars  were  studied  last,  hence 
the  letters  when  arranged  to  show  the  order  of  evolution  do 
not  follow  the  alphabetical  order.  However,  this  does  not 
cause  any  particular  inconvenience. 

Before  giving  the  letters,  it  is  desirable  to  state  on  what 
basis  the  new  grouping  was  made.  Miss  Cannon  selected  sev- 
eral groups  of  lines  which  act  together  as  described  above  and 
which  vary  in  intensity  in  passing  from  one  type  to  another. 
They  are  briefly  as  follows:  (1)  the  hydrogen  series,  which 
includes  not  only  the  four  lines  in  the  visible  spectrum  but 
an  extension  of  them  into  the  ultra  violet;  (2)  a  secondary 
hydrogen  series,  known  as  the  Pickering  series,  found  in  the 


32  THE  STUDY  OF  VARIABLE  STARS 

spectra  of  certain  stars,  and  not  known  terrestrially;  (3)  the 
Orion  lines,  which  include  helium,  a  few  lines  each  of  nitrogen, 
oxygen,  silicon,  etc.,  with  some  additional  strong  lines  due  to 
unknown  substances;  (4)  the  calcium  lines  H  and  K,  which 
are  so  intense  in  the  solar  spectrum;  (5)  solar  lines,  including 
lines  from  many  metals;  (6)  group  G;  (7)  a  group  of  bright 
bands  of  unknown  origin. 

The  classification  depends  entirely  upon  the  presence  and 
varying  intensities  of  these  groups  of  lines  and  upon  the  gen- 
eral absorption  in  the  spectrum.  It  is  briefly  described  below, 
the  classes  being  arranged  in  the  apparent  order  of  develop- 
ment. There  are  subdivisions  intermediate  between  the  sep- 
arate letters  which  are  indicated  by  letters  or  numbers  on 
the  scale  of  ten.  The  main  divisions  in  order  of  evolution  are 
O,  B,  A,  F,  G,  K,  M.  The  correspondence  with  Secchi's  types 
is  as  follows:  O,  Pickering's  fifth  type;  A,  B,  Secchi  I;  F,  I-II; 
G,  II;  K,  II-III;  M,  III.  Secchi's  IV  is  N,  but  there  is  no 
connection  between  it  and  type  M,  hence  it  does  not  belong 
to  the  series.  The  detailed  description  of  each  class  will  now 
be  given. 

Oa-Oc  are  bright-line  stars.  They  contain  one  or  both  of 
the  bands  (7)  just  mentioned  and  a  few  bright  lines,  princi- 
pally of  the  two  hydrogen  series.  Od  has  the  bright  bands,  and 
dark  lines  of  both  hydrogen  series  of  strong  intensity.  Oe  is 
similar  to  this  with  more  dark  lines.  Following  this  O  group  is 
a  subdivision  which  is  plainly  intermediate  between  the  O 
and  B  groups,  as  in  it  the  bright  bands  have  disappeared,  leav- 
ing only  dark  lines,  of  which  the  hydrogen  and  helium  lines 
have  about  the  same  intensity  as  in  class  B.  This  is  called 
Oe5B. 

Class  B  has  ten  subdivisions  designated  as  B,  BlA,  B2A, 
etc.,  which  are  often  abbreviated  as  BO,  Bl,  B2,  etc.  They 
are  marked  by  the  diminishing  intensity  and  early  disappear- 
ance of  the  secondary  hydrogen  spectrum,  by  the  increasing 
strength  of  the  usual  hydrogen  series,  by  the  diminution  of  the 
helium  and  other  Orion  lines,  and  toward  the  end  of  the  group 


Plate  II 

TYPICAL   STELLAR  SPECTRA 


ORIONIS 


a  CAN.  MAJ 


a  CARINAE 


a  CAN.  MIN. 


a  AURIGAE 


a  Bobxis 


a  ORIONIS 


INTRODUCTORY  33 

by  the  entrance  of  faint  solar  lines,  so  that  in  B8  and  B9  solar 
and  Orion  lines  are  intermingled. 

Class  A  has  fewer  subdivisions  than  Class  B;  these  are  A2F, 
A3,  and  A5.  The  typical  star  of  division  A  is  Sirius,  which  is 
marked  by  very  intense  hydrogen  lines.  These  extend  in  some 
stars  as  far  as  Her.  The  helium  lines  are  entirely  gone;  the  solar 
lines  are  present  and  increase  in  intensity  toward  Class  F. 
The  calcium  line  K,  which  is  faint,  increases  also  in  intens- 
ity, until  it  surpasses  H8.  The  calcium  line  H  is  so  nearly 
coincident  with  He  that  the  line  observed  is  a  combination  of 
the  two.  The  hydrogen  lines  decrease  in  intensity  as  the  class 
advances. 

Class  F  represents  spectra  in  which  the  bands  of  H  and  K, 
calcium,  are  the  most  conspicuous  features,  and  the  hydrogen 
lines  are  more  intense  than  any  solar  lines.  The  gradations 
between  this  class  and  the  next  are  F2G,  F5  and  F8.  Certain 
lines  in  band  G  appear  and  increase  in  intensity,  but  the  band 
is  not  well  marked. 

Class  G  contains  stars  with  spectra  in  which  the  lines  H 
and  K  of  calcium  and  the  band  G  are  the  most  prominent 
features,  while  the  hydrogen  lines  are  still  as  intense  as  any 
single  solar  lines. 

Class  K  represents  spectra  of  the  advanced  solar  type,  in 
which  the  bands  H  and  K,  G  and  the  calcium  line  4227  or  g 
are  the  most  conspicuous  features,  the  end  of  short  wave-length 
is  faint,  and  the  distribution  of  light  in  the  spectrum  is  not 
uniform.  The  hydrogen  lines  are  fainter.  Intermediate  between 
G  and  K  is  G5K,  and  between  K  and  M  are  K2  and  K5. 

Class  M  includes  the  banded  type.  Its  two  divisions  are 
Ma  and  Mb.  A  third  subdivision,  Md,  includes  stars  of  this 
type  which  occasionally  have  bright  hydrogen  lines. 

In  tracing  the  development  of  the  classes  from  one  to  the 
next,  the  progressive  changes  may  be  described  as  follows: 
There  first  appear  broad  hazy  bright  bands  (7)  which  finally 
disappear.  Simultaneous  with  them  are  bright  hydrogen  lines 
of  both  series  which  become  narrower  and  finally  give  place 


34  THE  STUDY  OF  VARIABLE  STARS 

to  dark  hydrogen  lines  of  both  series.  The  principal  series  in- 
creases in  intensity,  reaching  its  maximum  in  type  A,  after 
which  it  diminishes,  becoming  less  and  less  conspicuous  until 
in  class  M  the  lines  are  fainter  than  many  of  the  solar  lines. 
The  second  series  of  hydrogen  lines  reaches  its  maximum  in- 
tensity in  Od  and  then  quickly  disappears.  As  the  bright 
bands  (7)  disappear,  the  Orion  lines  appear,  increase  in  in- 
tensity, reaching  a  maximum  in  B2  and  B3,  then  diminish 
and  in  A  are  hardly  visible.  The  calcium  line  K  appears  in 
Class  A,  and  increases  in  intensity  until  together  with  H  it 
dominates  the  spectrum.  The  other  solar  lines  which  appear 
faintly  in  type  A  become  more  and  more  strengthened,  par- 
ticularly band  G  and  the  calcium  line  4227.  Finally  the  spec- 
trum becomes  banded. 

CONNECTION  BETWEEN   SPECTRAL  TYPE  AND   TYPE   OF 
VARIATION 

CLASS  I.  Temporary  stars  have  always  the  same  type  of 
spectrum,  which  consists  of  bright  and  dark  bands  of  hydro- 
gen and  helium  side  by  side.  In  addition  are  usually  seen  the 
D  lines  of  sodium  and  the  H  and  K  lines  of  calcium.  The  most 
striking  characteristic  of  the  spectrum  is  the  great  displace- 
ment of  the  bands,  the  dark  ones  being  shifted  toward  the 
violet  end  of  the  spectrum  and  the  bright  bands  toward  the 
red.  Fine  lines  may  also  be  detected  with  very  bright  novae, 
and  many  changes  take  place  in  the  spectra.  However,  this 
subject  has  been  treated  so  fully  elsewhere  that  we  need  not 
repeat  the  facts  here.  It  is  sufficient  for  the  present  purpose 
to  state  that  the  spectrum  of  dark  and  bright  bands,  with  the 
displacement  described  above,  always  is  indicative  of  a  new 
star,  and  that  whenever  such  a  spectrum  has  been  found  on  a 
photographic  plate,  and  the  photometric  history  of  the  star 
has  been  investigated,  it  has  been  proved  to  be  a  new  star. 

It  should  be  added,  that  in  two  cases  a  new  star  has  been 
caught  early  enough  in  its  history  to  show  a  different  type  of 
spectrum,  for  a  brief  time  only,  as  the  typical  aspect  has  devel- 


INTRODUCTORY  35 

oped  very  quickly.  These  stars  are  Nova  Persei,  1901,  and 
Nova  Geminorum,  No.  2. 

CLASS  II.  Long  period  variables.  The  spectrum  is  with  strik- 
ing uniformity  that  of  Secchi's  type  III  or  Harvard  M,  but 
showing  bright  hydrogen  lines  at  maximum,  whenever  it  has 
been  photographed  at  that  time.  This  fact  is  made  use  of  in 
the  discovery  of  long  period  variables,  just  as  the  banded  spec- 
trum of  bright  and  dark  bands  is  made  a  test  for  temporary 
stars,  and  when  a  third  type  spectrum  is  found  with  bright 
hydrogen  lines,  the  star  is  marked  as  a  suspected  variable  and 
subjected  to  further  investigation.  The  spectrum  is  also  made 
use  of  in  separating  variables  of  Class  II  from  those  of  Class 
IV,  for  as  stated  earlier,  the  dividing  line  is  not  based  entirely 
on  length  of  period.  So  far,  the  variable  in  Class  IV  having 
the  longest  period  is  SS  Geminorum  with  a  period  of  45  days, 
and  the  variable  of  Class  II  having  the  shortest  period  is  SZ 
Cassiopeiae  with  a  period  of  50  days.  If  a  variable  were  to  be 
found  with  a  period  of  40  days  and  a  spectrum  of  type  III 
it  would  be  placed  in  Class  II,  not  in  Class  IV.  A  few  vari- 
ables of  Class  II  have  continuous  spectra  or  the  spectrum  of 
Class  N. 

CLASS  III.  Irregular  variables.  With  few  exceptions  these 
have  spectra  of  Class  M  or  N  and  hence  are  supposed  to  be  in 
a  very  late  stage  of  evolution.  Their  irregularity  is  thought  to 
be  due  to  the  fact  that  the  forces  which  cause  the  variation 
are  dying  out.  Hence  no  star  would  be  placed  in  this  class, 
which  has  a  spectrum  of  a  much  earlier  type,  but  if  it  ap- 
peared to  be  irregular,  it  would  rather  be  classed  as  unknown. 
One  well-known  illustration  is  u  Herculis,  which  was  men- 
tioned in  the  section  on  the  classification  of  variables.  This 
star  had  for  several  years  been  known  to  vary  with  a  small 
range,  and  was  called  irregular,  but  later  it  was  found  to  be 
of  the  /3  Lyrae  type,  having  a  period  of  2.05  days. 

CLASS  IV.  Short  period  variables.  The  Cepheid  stars  of  this 
type  have  spectra  mainly  of  Class  G  or  F.  Stars  belonging 
to  the  /3  Lyrae  subdivision  have  spectra  of  an  earlier  type, 


36  THE  STUDY  OF  VARIABLE  STARS 

B  or  A  predominating.  0  Lyrae  itself  is  one  of  the  most  in- 
teresting and  baffling  stars  in  the  sky.  Its  spectrum  is  of  the 
B  type,  but  it  has  bright  and  dark  lines  of  the  same  elements, 
particularly  hydrogen  and  helium. 

CLASS  V.  Algol  type.  The  stars  in  this  group  also  have  an 
early  spectral  type,  A  and  F  predominating. 

The  data  upon  which  the  last  section  has  been  founded  have 
been  taken  largely  from  the  tables  in  H.C.O.,  Annals,  vols. 
55  and  56,  though  some  material  has  been  found  in  the  cur- 
rent periodicals. 

It  has  been  the  purpose  in  this  chapter  to  give  a  general 
survey  of  the  facts  necessary  for  a  reasonable  understanding 
of  the  many  technical  references  that  must  be  made  in  the 
succeeding  chapters.  Many  points  have  been  touched  upon 
which  will  be  treated  fully  at  a  later  time,  but  a  preliminary 
knowledge  of  the  meaning  of  stellar  variation,  of  the  different 
classes  of  curves,  of  the  types  of  stellar  spectra  and  their  rela- 
tion to  the  different  classes  of  variables  was  considered  essen- 
tial by  the  writer. 

[NOTE.  —  Among  the  seven  sets  of  lines  which  Miss  Cannon  de- 
scribes in  giving  the  basis  of  the  Harvard  classification  of  stellar 
spectra,  there  are  two  which  merit  special  attention  on  account  of 
some  recent  investigations  concerning  them.  These  are  (2),  the  Pick- 
ering series  of  hydrogen  found  in  £  Puppis,  and  (7),  the  bright  bands 
of  unknown  origin.  Both  had  been  ascribed  to  hydrogen,  but  it  was 
thought  that  they  were  produced  by  that  element  under  conditions 
which  could  not  be  duplicated  terrestrially.  Quite  recently,  Fowler,1 
at  the  Solar  Physics  Laboratory  at  South  Kensington  has  obtained 
the  bright  bands  (7)  and  a  few  lines  of  the  series  (2)  by  passing  a 
strong  condensed  discharge  through  a  mixture  of  hydrogen  and  helium 
in  a  vacuum  tube.  It  is  not  possible  to  give  here  anything  further  in 
regard  to  the  experiments  except  to  quote  his  final  words,  which  are 
extremely  interesting  and  satisfying  to  the  astronomer.  "The  pro- 
duction of  the  new  lines  gives  a  further  indication  of  the  probability 
that  there  are  no  special  kinds  of  matter  in  celestial  bodies,  and  that 
most,  if  not  all,  of  the  celestial  spectra  are  well  within  range  of  labora- 
tory experiments." 

*  Mon.  Not.  R.A.S.,  73,  62. 


INTRODUCTORY  37 

Whether  the  lines  belong  to  hydrogen  or  helium  is  still  a  moot  ques- 
tion. Theoretical  considerations  connected  with  series  of  lines  in 
spectra  seem  to  point  to  helium  as  being  their  source.  In  fact,  one 
of  the  lines  was  found  in  a  tube  which  contained  helium1  but  not 
hydrogen.  The  matter  cannot  yet  be  considered  as  settled  experi- 
mentally.] 

1  Evans,  Nature,  Sept.  4,  1913,  p.  5. 


CHAPTER  II 

STAR  CHARTS  FOR  GENERAL  USE 

STAB  charts  intended  for  general  use  by  astronomers  may 
be  divided  into  two  classes,  those  which  are  suited  for  iden- 
tifying the  lucid  stars  and  those  which  are  prepared  for.  tele- 
scopic work.  In  the  first  class  may  be  found  the  well-known 
atlases  of  Heis,  Schurig,  Klein,  and  Upton,  and  in  the  second 
the  great  Durchmusterung  of  Argelander  (which  deserves  to 
stand  in  a  class  by  itself),  and  printed  charts  covering  special 
regions  of  the  sky  such  as  the  I>aris  Ecliptic  Charts. 

The  first  modern  set  of  maps  prepared  for  convenient  com- 
parison with  the  sky  were  from  the  hand  of  Argelander,  who 
published  in  1843  his  atlas  which  he  called  the  Uranometria 
Nova.1  As  the  name  implies,  its  chief  purpose  was  to  give  the 
correct  magnitudes  of  the  stars,  but  this  was  not  its  only  one, 
as  the  introductory  sentences  state:  "All  of  the  star  charts 
which  we  possess  up  to  the  present  time,  are  lacking  in  two 
important  respects;  —  the  magnitudes  of  the  stars  depend 
upon  estimations  which  have  been  made  at  the  telescope  by 
astronomers  while  in  the  process  of  determining  their  positions, 
and  are  for  the  most  part  quite  erroneous.  There  are  also  lack- 
ing quite  a  number  of  the  brighter  stars  while  many  of  the 
fainter  ones  are  included.  Both  together  often  change  the 
constellations  so  completely  that  one  can  scarcely  follow  the 
charts,  particularly  in  regions  which  are  poor  in  the  brighter 
stars.  It  is  the  purpose  of  the  present  charts  to  supply  this 
want  as  far  as  possible,  for  those  in  middle  Europe  who  wish 
to  observe  stars  visible  to  the  naked  eye." 

These  famous  charts,  which  are  now  out  of  print  and  have 

1  Fr.  Argelander.  Neue  Uranometrie.  Darstellung  der  in  mittleren  Europa 
mit  blossen  Augen  sichtbaren  Sterne  nach  ihren  wahren,  unmittelbar  vom 
Himmel  entnommenen  GrOssen.  Berlin,  1843. 


STAR  CHARTS  FOR  GENERAL  USE  39 

been  superseded  by  other  more  modern  ones,  are  worth  know- 
ing about,  because  they  were  the  basis  of  so  much  later  work. 
In  appearance  they  resemble  the  maps  of  Heis,  and  are  ac- 
companied by  a  book  containing  an  account  of  the  method  of 
forming  the  charts  and  a  catalogue  of  the  stars  found  on  them. 
Argelander's  method  of  determining  the  star  magnitudes, 
which  is  the  most  important  part  of  the  work,  will  be  de- 
scribed in  Chapter  V. 

The  atlas  of  Heis,1  which  was  modeled  after  the  Nova  Urano- 
metria  of  Argelander,  consists  of  thirteen  plates  representing 
the  sky  as  far  south  as  —30°  declination,  and  including  stars 
as  faint  as  6+  magnitude.  The  Milky  Way  is  very  carefully 
represented  on  this  atlas  in  five  degrees  of  density.  It  also  con- 
tains the  figures  of  the  constellations  faintly  outlined  in  red, 
artistically  drawn,  being  copied  from  the  Farnese  globe  in  the 
Naples  Museum,  a  cut  of  which  appears  on  the  title-page.  The 
magnitudes  of  the  stars  are  represented  by  different  symbols, 
as  is  customary.  Accompanying  the  atlas  is  a  catalogue  with  a 
Latin  introduction  in  which  the  author  pays  a  tribute  to  the 
vir  illustrissimus  Argelander,  with  whom  he  had  been  associ- 
ated in  the  study  of  variable  stars.  The  magnitudes  were 
deduced  from  Heis's  own  observations,  which  extended  over  a 
period  of  twenty-seven  years.  As  he  had  remarkably  clear 
eyesight  he  was  able  to  include  many  stars  not  ordinarily  seen, 
so  that  on  his  maps  there  are  5421  stars,  being  2153  more  than 
Argelander  represented  in  his  Uranometria.  The  atlas  was 
published  in  1872.  Unfortunately,  it  is  of  a  shape  which  ren- 
ders it  a  little  unhandy  for  common  use,  and  its  expense  re- 
duces the  demand  for  it.  Furthermore,  the  right  ascensions 
and  declinations  are  for  the  year  1855,  the  former  being  ex- 
pressed in  degrees  instead  of  hours,  which  makes  it  incon- 
venient for  ready  use  at  the  present  time. 

Schurig's  atlas,  which  is  intended  for  the  same  purpose,  is 
arranged  much  more  conveniently,  is  less  than  a  quarter  the 
price,  and  a  second  edition  published  in  1909  contains  many 
1  Eduard  Heis,  Atlas  Coelestis  Novus.  Coeln,  1872. 


40  THE  STUDY  OF  VARIABLE  STARS 

desirable  improvements.  The  principal  facts  stated  in  the 
preface  may  be  summarized  as  follows.  The  positions  of  the 
stars  are  for  the  equinox  1925.0.  Their  magnitudes  are  taken 
from  the  Potsdam  Photometric  Durchmusterung  and  in  the 
southern  zones  from  the  Harvard  Photometry.  Many  nebulae 
and  star  clusters  are  included,  also  variable  stars  and  doubles. 
There  are  symbols  for  the  magnitudes  and  thirds  from  1  to 
6j,  or  17  in  all.  The  Milky  Way  is  represented  in  several  de- 
grees of  brightness.  The  stars  are  printed  in  black,  their 
names  and  the  boundary  lines  of  the  constellations  in  red,  so 
that  the  maps  are  well  adapted  for  use  at  night  with  artificial 
light.  There  are  eight  maps  in  all,  which  cover  the  entire 
heavens. 

While  the  magnitudes  on  such  maps  are  not  to  be  consid- 
ered in  any  sense  as  definitive,  it  is  an  added  convenience  to 
have  them  assigned  with  care  so  that  they  can  be  used  by  a 
beginner  in  testing  his  power  to  distinguish  different  degrees 
of  brightness.  Their  principal  use  is  for  the  observer  with  a 
portable  telescope  which  has  no  circles  for  setting.  He  must 
be  able  to  connect  a  prominent  star  in  the  sky,  through  some 
definite  configuration  which  can  be  picked  out  on  Schurig's 
atlas,  with  the  group  in  the  field  of  his  telescope  containing 
the  variable.  Further  discussion  of  this  point  will  be  given 
in  the  chapter  entitled  "Hints  for  Observers." 

Upton's  Star  Atlas  is  intended  to  serve  the  same  purpose 
as  the  others.  It  has  the  advantage  of  being  published  in 
America  and  hence  is  more  easily  obtainable  than  Heis  or 
Schurig.  Only  whole  magnitudes  are  represented.  There  are 
six  maps  covering  the  entire  heavens. 

The  Bonner  Durchmusterung  is  the  chief  member  of  the 
second  division  of  maps,  and  contains  324,000  stars  includ- 
ing those  as  faint  as  the  ninth  or  tenth  magnitude.  It  is  in 
two  distinct  parts;  —  the  northern  Durchmusterung,  prepared 
by  Argelander,  which  extends  from  the  north  pole  to  declina- 
tion —2°,  and  the  continuation  of  it,  which  was  completed  by 
Schonfeld,  and  extends  from  —2°  to  —23°  declination.  Since 


STAR  CHARTS  FOR  GENERAL  USE  41 

the  Banner  Durchmusterung  is  one  of  the  most  valuable  pieces 
of  astronomical  work  ever  executed,  it  has  seemed  worth  while 
to  the  author  to  give  a  somewhat  detailed  account  of  its  con- 
ception and  formation.  It  consists  of  both  star  catalogues 
and  charts,  and  is  of  such  great  value  that  when  the  supply 
became  exhausted  many  years  ago  the  question  of  issuing  a 
second  edition  was  seriously  agitated.  This  was  finally  ac- 
complished and  new  charts  were  published  in  1899,  being 
dedicated  to  Argelander  on  the  hundredth  anniversary  of  his 
birth,  which  occurred  on  March  22,  1799.  The  reprint  of  the 
catalogue  was  published  in  1903.  The  description  of  the  forma- 
tion of  the  catalogue  and  charts  is  taken  from  Argelander's 
introduction  to  the  first  edition,  the  dates  of  which  are  1859 
and  1863. 

The  idea  of  making  an  extensive  chart  of  the  heavens  was 
first  suggested  by  Bessel  in  an  article  in  the  Astronomische 
Nachrichten1  for  1822,  wherein  he  calls  attention  to  the  His- 
toire  Celeste  of  Lalande,  a  catalogue  containing  50,000  stars 
down  to  the  eighth  magnitude,  for  the  epoch  1800.  He  adds 
that  this  should  be  extended  so  that  there  may  be  a  complete 
catalogue,  with  charts,  of  all  stars  within  certain  limits  down 
to  the  ninth  magnitude,  the  principal  purpose  being  to  assist 
in  the  discovery  of  new  minor  planets.  He  did  not  think  it 
necessary  that  all  of  the  star  places  should  be  determined  by 
meridian  observations,  but  suggested  that  as  many  as  possible 
be  located  in  this  way  and  that  the  others  be  inserted  by  eye 
estimates  on  the  charts.  Bessel  entered  upon  the  execution  of 
this  scheme  with  the  assistance  of  Argelander,  then  only 
twenty-two  years  old,  not  with  the  expectation  of  completing 
the  survey,  but  for  the  purpose  of  trying  the  plan  and  seeing 
how  easily  it  could  be  carried  out.  As  a  result  of  his  experience 
he  came  to  the  conclusion  that  he  would  be  unable  to  complete 
it  himself,  and  asked  the  co-operation  of  other  astronomers. 

In  1825  he  again  wrote  to  the  Nachrichten,2  this  time  making 
quite  definite  propositions  and  begging  other  astronomers  to 
i  A.N.,  17.  2  A.N.,  88. 


42  THE  STUDY  OF  VARIABLE  STARS 

join  him  in  the  work.  His  plea  was  the  more  urgent  since  his 
accomplished  assistant,  Argelander,  had  left  Konigsberg  to 
become  the  director  of  the  observatory  at  Abo,  and  he  him- 
self was  engaged  in  other  work.  A  copy  of  his  preliminary 
chart  accompanied  his  article. 

It  was  this  plan  of  BessePs,  so  Argelander  states  in  the  in- 
troduction to  his  charts,  that  induced  him  to  undertake  the 
Durchmusterung.  The  making  of  the  catalogue  necessarily 
came  first.  He  desired  at  the  outset  to  have  the  charts  extend 
as  far  south  as  the  tropic  of  Capricorn  or  25°  south  declination, 
but  the  atmospheric  conditions  at  Bonn  did  not  permit  of 
this.  He  then  limited  himself  to  the  northern  heavens,  but 
included  the  .zone  of  —0°  to  —2°  in  order  to  connect  with  similar 
maps,  which  would  be  made,  so  he  hoped,  by  observers  in  the 
southern  hemisphere.  He  intended  the  positions  to  be  so  ac- 
curate that  the  error  would  not  exceed  one  minute  in  either 
co-ordinate,  and  that  thus  each  star  on  the  chart  would  easily 
be  found  again  with  a  meridian  instrument.  All  large  errors 
in  previous  star  catalogues  should  be  carefully  looked  for  and 
eliminated.  He  preferred  not  to  follow  the  method  previ- 
ously suggested  of  plotting  the  positions  of  stars  from  the 
catalogues  already  known  and  inserting  other  stars  by  eye 
estimates.  He  gives  several  reasons  for  this,  the  chief  one 
being  that  the  method  of  observation  was  too  exacting  on  the 
eye  of  the  observer,  owing  to  the  constant  changing  of  the 
illumination  in  looking  from  the  lighted  chart  to  the  dark  field 
of  the  telescope.  Furthermore,  the  resulting  positions  of  the 
stars  would  not  be  accurate  enough.  He  determined,  then, 
to  obtain  the  positions  of  all  the  stars  with  a  degree  of  accur- 
acy which  should  be  equal  and  suited  to  the  purpose  before 
him.  After  some  experimentation  he  adopted  the  following 
method  of  observation. 

A  Fraunhofer  comet  seeker  of  34  lines  or  3  inches  aperture 
and  two  feet  focal  length,  furnished  with  an  eyepiece  magnify- 
ing ten  diameters,  was  installed  in  the  south  tower  of  the 
observatory.  A  special  eyepiece  was  constructed  for  it,  in  the 


STAR  CHARTS  FOR  GENERAL  USE  43 

focus  of  which  was  placed  a  semi-circular  piece  of  thin  glass 
oriented  in  such  a  way  that  the  straight  edge  or  diameter 
formed  an  hour  circle.  In  the  telescope  it  appeared  as  a  thin 
dark  line  which  in  the  complete  absence  of  artificial  light  could 
be  seen  by  the  faint  illumination  due  to  starlight  alone.  Per- 
pendicular to  it  was  drawn  a  radius  at  the  middle  point,  and 
parallel  to  this  on  either  side  at  intervals  of  7'  were  drawn  ten 
shorter  marks,  every  third  one  being  a  little  longer,  to  allow  of 
easy  discernment.  These  were  not  readily  seen  in  the  dark 
field,  and  were  made  visible  by  being  drawn  with  thick  black 
oil  paint,  which  made  them  rather  broad  and  hence  likely  to 
cause  some  error,  but  one  which  was  considered  to  be  less  than 
the  error  of  observation. 

The  observer,  who  was  called  "A,"  assumed  as  comfortable 
an  attitude  as  possible  in  placing  his  eye  at  the  telescope,  no 
change  in  his  position  being  required,  as  the  stars  were  always 
observed  in  narrow  zones.  There  was  no  artificial  light  in  the 
room,  and  the  eye  was  protected  against  the  light  from  the  sky 
by  means  of  a  dark  cardboard  screen  which  surrounded  the  eye 
end.  Under  the  observing  room  was  another  room  in  which 
was  a  sidereal  clock  before  which  the  assistant  "B"  was  seated. 
Only  a  simple  board  floor  separated  the  two  rooms,  so  that  a 
sound  could  easily  be  heard  from  one  to  the  other.  When  the 
work  was  ready  to  begin,  the  telescope  was  set  for  the  proper 
declination  and  right  ascension;  the  observer  seated  himself, 
and  the  assistant  withdrew  from  the  room  carrying  the  arti- 
ficial light  with  him.  On  a  table  close  to  the  observer  was 
placed  a  pile  of  papers,  each  fitted  with  a  rack  dividing  it  into 
five  vertical  columns,  so  that  the  observer  "A"  could  pick  it 
up  in  the  dark,  and  write  in  the  columns,  running  his  hand 
down  the  forms  without  taking  his  eye  from  the  telescope  or 
seeing  the  paper.  Another  compartment  of  the  table  was 
reserved  to  place  them  in  when  finished. 

In  making  the  observation  "A"  called  out  to  "B"  the  mag- 
nitude of  the  star  and  the  instant  that  it  crossed  the  hour 
circle.  He  himself  wrote  down  the  declination  north  or  south 


44  THE  STUDY  OF  VARIABLE  STARS 

of  the  mid-line  of  the  field,  and  made  any  other  necessary  notes. 
When  a  paper  was  finished,  he  gave  notice  of  the  fact  to  "B," 
who  drew  a  line  across  his  own  record,  and  the  same  was  done 
when  anything  happened  to  disturb  the  observer  and  make 
him  pause.  This  kind  of  observation  proved  to  be  so  exacting 
that  the  observers  could  work  at  it  for  about  an  hour,  or  at 
most  an  hour  and  a  quarter,  when  it  became  necessary  to 
change.  When  "A"  gave  the  signal  for  stopping,  "B"  pulled  a 
bell  which  rang  in  another  room  where  two  other  observers 
were  waiting  to  take  their  places.  While  waiting  for  them  to 
arrive,  "A"  read  the  circles.  The  first  pair  then  retired  to  the 
other  work  room,  went  over  their  two  records  together,  to  see 
that  everything  was  in  agreement,  and  to  clear  up  misunder- 
standings if  possible,  while  their  memories  were  still  fresh  from 
the  work. 

This  brief  description  of  the  method  pursued  by  Argelander 
and  his  assistants  shows  how  it  was  possible  for  him  to  achieve 
such  an  enormous  piece  of  work  in  so  short  a  time.  The  first 
section  of  the  catalogue,  containing  110,985  stars  between  the 
limits  —2°  and  +20°  declination,  was  published  in  1859;  the 
second,  containing  105,075  stars,  between  20°  and  40°,  in  1861; 
the  third,  containing  108,129  stars,  between  40°  and  90°,  in 
1862.  The  epoch  is  1855.  The  charts  were  finished  in  1863.  It 
is  of  course  impossible  that  in  such  a  piece  of  work  there  should 
be  no  errors.  Argelander  himself  refers  to  the  probability  of 
their  existence,  but  states  that  as  the  papers  were  arranged  in 
the  most  complete  order,  and  carefully  preserved  in  the  library 
of  the  observatory,  he  hopes  that  they  will  be  freely  used  for 
reference  in  all  doubtful  cases. 

The  astronomer  of  to-day  is  fully  aware  how  this  wish  has 
been  fulfilled,  and  one  frequently  sees  in  the  Astronomische 
Nachrichten  letters  from  Professor  Kiistner,  the  present  director 
of  the  observatory  at  Bonn,  written  in  response  to  inquiries 
made  by  observers  whose  results  may  differ  from  those  con- 
tained in  the  Durchmusterung,  in  which  he  quotes  the  original 
records.  An  example  may  be  found  in  the  Astronomische 


STAR  CHARTS  FOR  GENERAL  USE     45 

Nachrichten  4383,  regarding  the  magnitudes  of  a  star,  and 
another  in  the  Astronomische  Nachrichten  4386,  in  which  there 
was  some  ambiguity  about  the  positions  of  two  adjacent  stars. 

Mention  should  be  made  of  the  limiting  magnitude  set  by 
Argelander  in  his  work.  It  was  his  intention  to  include  all  the 
stars  down  to  the  ninth  magnitude  in  the  region  charted,  all 
the  brighter  stars  of  the  class  9.10,  and  as  many  more  of  this 
class  as  the  circumstances  would  permit.  That  is  to  say,  in  a 
region  where  the  stars  were  sparsely  scattered,  more  of  the 
fainter  ones  would  be  observed,  but  in  richer  regions,  such  as 
the  Milky  Way,  perhaps  not  all  even  of  the  brighter  ones  of 
this  magnitude  would  be  included.  As  a  result,  the  Durchmus- 
terung  magnitudes  below  the  ninth  are  not  reliable. 

The  arrangement  of  the  stars  in  the  catalogue  may  now  be 
explained,  and  at  this  juncture  the  author  wishes  to  state  that 
the  Durchmusterung  is  thus  fully  described  not  only  on  account 
of  its  universal  interest,  but  because  observers  of  variable  stars 
need  to  make  frequent  use  of  it  in  preparing  their  star  maps, 
and  hence  will  find  directions  for  its  use  very  convenient.  The 
description  should  be  used  in  connection  with  the  catalogue. 

The  catalogue  is  divided  into  zones  one  degree  wide.  Each 
page  has  five  similar  columns  in  which  the  stars  are  separated 
into  groups  of  ten.  The  current  numbers  comprised  in  any  one 
column  are  printed  at  the  top  of  it,  and  hence  need  not  be  given 
for  the  individual  stars,  but  may  readily  be  found  by  counting 
down  from  the  top  of  the  column.  The  division  into  groups  is  to 
facilitate  the  identification  of  a  star. 

For  example,  on  the  first  page  of  the  catalogue  is  given  the 
declination  of  the  zone,  —1°,  and  at  the  heads  of  the  columns 
stand  the  numbers  1-40,  41-80,  81-120,  121-160,  161-200. 
Number  136  in  this  zone  will  be  found  in  the  fourth  column,  in 
the  second  group,  and  will  be  the  sixth  star  in  the  group.  Stars 
in  this  catalogue  are  usually  designated  by  the  declination  of 
the  zone  followed  by  the  number  in  the  zone,  the  number  being 
preceded  by  the  letters  BD;  for  example,  the  star  just  mentioned 
is  BD  —1°  136.  The  one  referred  to  in  Astronomische  Nach- 


46  THE  STUDY  OF  VARIABLE  STARS 

richten  4383  is  BD  +34°  4598.  Under  the  current  numbers 
stands  the  hour  of  right  ascension.  Each  of  the  five  columns  of 
stars  on  a  page  has  itself  four  columns  of  numbers.  The  first 
gives  the  magnitude,  the  second  the  minutes  and  seconds  of 
right  ascension,  which,  contrary  to  the  modern  custom,  are 
indicated  by  the  strokes '  and "  instead  of  the  letters  m  and  s. 
The  right  ascension  is  given  to  tenths  and  not  hundredths  of  a 
second.  The  next  column  gives  the  minutes  and  tenths  of 
declination,  the  degrees  standing  at  the  head  of  the  column. 
The  fourth  column  gives  the  references  to  other  star  catalogues, 
each  one  being  represented  by  its  particular  abbreviation.  At 
the  head  of  each  page  in  bold  type  are  the  degree  of  the  zone 
and  the  hour  of  right  ascension.  For  example  the  data  concern- 
ing star  BD  -1°  136  are  9.3  mg.,  Oh  56m  9.2s,  -1°  10'.9. 
No  letter  of  reference  is  given,  hence  the  star  cannot  be  found 
in  any  other  catalogue. 

Since  the  charts  also  are  in  very  general  use,  a  description  of 
them  will  probably  be  of  service  to  the  beginner.  They  are 
printed  on  sheets  30  x  21  inches  and  are  in  four  zones  with  one 
circular  map  for  the  polar  region,  numbering  forty  sheets  in  all. 
The  first  tier  covers  the  region  from  —2°  to  +20°,  or  an  extent 
of  22°  in  declination;  the  second  19°  to  41°,  the  third  40°  to  61°, 
the  fourth  60°  to  80°  and  the  last  one  79°  to  90°.  Each  chart  is 
covered  with  a  network  of  lines  which  are  one  degree  apart  in 
declination  and  four  minutes  of  time  in  right  ascension  on  the 
first  three  tiers  of  charts,  and  eight  minutes  on  the  fourth.  On 
the  first  tier,  the  network  is  square,  the  side  of  each  square 
being  20  mm.  On  the  other  tiers  the  hour  circles  are  closer 
together  and  converge  according  to  the  cosine  of  the  declina- 
tion. In  right  ascension  they  overlap  one  tier,  or  four  minutes. 

A  particular  use  of  the  charts  is  for  locating  a  star  and  giving 
the  configuration  of  the  surrounding  region.  In  order  to  locate 
a  star  we  must  first  find  the  square  in  which  it  occurs.  The 
sides  of  the  square  being  four  minutes  apart  in  right  ascension, 
the  star  will  be  included  between  two  hour  circles  which  are 
multiples  of  four  minutes.  After  the  square  has  been  found,  the 


STAR  CHARTS  FOR  GENERAL  USE 


47 


Figure  15 


star  may  be  located  by  proportioning  for  the  difference  in  right 
ascension  and  declination,  or  more  easily  by  counting  directly 
from  the  preceding  side  of  the  square,  as  will  be  seen  from  the 
following  example. 

The  star  BD -1°  277, 
8.3  mg.,  Ih  26m  10.3s,  + 
1°  50'.3  lies  in  the  square 
between  1°  and  2°  declin- 
ation and  in  right  ascen- 
sion Ih,  between  24m  and 
28m.  On  referring  to  the 
catalogue,  we  find  that 
the  first  star  in  this  square 
has  declination  50'.6.  Fol- 
low the  stars  in  the  square 
in  order  with  a  pointer 
in  the  right  hand  and 
the  stars  in  the  catalogue 
with  another  pointer  in 
the  left  hand.  The  second 

star  in  the  square  has  declination  20'.2,  the  third  51'.6,  the 
fourth  52'.8,  the  fifth  17'.4,  the  sixth  17'.6,  and  the  seventh, 
which  is  the  one  desired,  50'.3. 

The  size  of  the  dot  shows  that  it  has  the  given  magnitude 
8.3.  After  many  years  of  experience  in  using  these  charts  for 
many  different  purposes,  the  writer  has  found  this  to  be  the 
quickest  way  to  locate  a  star  and  the  one  least  liable  to  error. 
If  it  should  happen  that  only  the  position  of  the  star  is  given 
and  the  catalogue  is  not  at  hand,  it  would  be  necessary  to  use 
the  first  method  and  to  locate  the  star  by  proportioning. 

On  the  charts  the  variables  which  were  known  at  the  time  of 
Argelander  are  marked  with  the  abbreviation  var.  The  size  of 
the  dot  represents  the  average  magnitude  at  maximum.  Double 
stars  are  indicated  by  a  line  drawn  under  the  star. 

The  work  was  begun  in  1852,  continuing  seven  years  and  one 
month,  and  during  this  time  1841  zones  were  observed  on  625 


SQUARE  FROM  DURCHMUSTERUNG 
CHART 


48  THE  STUDY  OF  VARIABLE  STARS 

nights,  12  of  them  by  Argelander  himself,  and  the  remainder 
by  his  assistants.  The  total  number  of  observations  is  1,650,000, 
which  belong  to  324,198  stars.  On  the  average  each  position 
depends  upon  2f  observations.  Argelander  pays  a  remarkable 
tribute  to  his  collaborators  in  the  work,  Schonf eld  and  Krueger, 
referring  to  them  as  follows:  — 

These  gentlemen  have  diligently  shared  the  work  with  me  for  many 
years.  With  the  exception  of  the  first  period,  I  have  given  over  to 
their  younger  and  stronger  eyes  alone  the  zone  observations  and  in 
all  the  remaining  labors  I  have  enjoyed  their  earnest  support.  They 
have  won  for  themselves  so  much  real  advantage  from  the  success  of 
the  work  that  I  can  consider  it  as  one  carried  out  by  them  and  myself 
in  common,  and  it  gives  me  the  greatest  pleasure  to  express  my 
warmest  thanks  for  the  judgment,  zeal  and  perseverance  with  which 
they  have  devoted  to  this  work  their  splendid  talents. 

As  stated  before,  the  second  edition  of  the  charts  was  pub- 
lished as  a  memorial  to  Argelander  on  the  one  hundredth  anni- 
versary of  his  birth,  the  work  being  undertaken  at  the  Bonn 
Observatory.  In  the  introduction  which  was  prepared  by 
Kustner,  the  Director  of  the  Observatory,  he  wrote  that  unfor- 
tunately the  stones  on  which  the  originals  were  lithographed 
had  not  been  preserved,  but  that  it  was  doubtful,  even  if  they 
had  been  kept,  whether  they  could  have  been  used  after  so 
long  a  lapse  of  time.  He  found  on  investigation  that  modern 
methods  of  reproduction  would  give  satisfactory  results  and 
the  photo-lithographic  process  was  employed  at  the  Imperial 
Printing  Bureau  in  Berlin.  Corrections  of  all  the  errors  known 
at  the  time  were  made,  many  of  which  were  contributed  by 
different  astronomers  who  had  used  the  charts  and  had  dis- 
covered mistakes,  and  the  entire  series  was  most  carefully 
revised. 

The  second  edition  of  the  catalogue  was  printed  in  1903  under 
the  direction  of  Professor  Kustner.  As  was  the  case  with  the 
charts,  the  original  was  excellently  reproduced  by  a  photo- 
graphic process. 

In  preparing  for  the  new  edition  it  was  possible  to  make 


STAR  CHARTS  FOR  GENERAL  USE  49 

many  necessary  corrections.  Changes  and  addenda  were  drawn 
in  with  the  lithographic  pen  but  in  somewhat  larger  and  more 
slanting  type,  in  order  to  be  recognized  easily.  Where  star 
places  or  letters  were  to  be  deleted,  this  was  indicated  by 
having  a  horizontal  stroke  drawn  through  them  but  in  such  a 
way  that  the  original  figures  could  plainly  be  read.  New  vari- 
able stars  were  indicated  by  the  syllable  var,  additional  stars 
were  added  by  being  placed  at  the  bottom  of  the  columns,  an 
asterisk  in  column  one  indicating  where  each  should  be  inserted 
in  the  catalogue.  At  the  bottom  of  each  column  is  given  ten 
years  precession  in  right  ascension  and  declination,  which  are 
to  be  used  in  carrying  the  position  from  the  date  of  the  catalogue 
to  any  desired  epoch. 

The  continuation  of  the  Durchmusterung  to  declination  —23° 
was  carried  out  by  Schonfeld,  who  was  Argelander's  successor 
at  the  Observatory  at  Bonn.  The  catalogue  was  published  in 
1886  and  the  charts  in  1887.  They  are  dedicated  to  the  memory 
of  Argelander,  whose  original  design  it  was  to  complete  the 
work  himself,  a  task  which  he  was  not  able  to  carry  out  on 
account  of  the  low  altitude  of  the  stars  and  the  small  aperture 
of  the  telescope.  He  did,  however,  observe  44  zones  south  of 
—2°.  Schonfeld,  also  finding  that  the  telescope  of  Argelander 
was  too  small  to  show  the  faint  stars  well,  used  a  six-inch  tele- 
scope with  a  higher  magnifying  power.  The  field  was  thus 
darker  and  hence  he  was  obliged  to  use  artificial  illumination 
to  see  the  lines  in  the  eyepiece.  He  extended  the  catalogue  to 
include  stars  of  magnitude  10.0.  In  form  it  is  similar  to  that 
of  Argelander,  though  the  charts  are  somewhat  different  in 
shape  since  they  cover  a  wider  difference  in  declination.  They 
are  also  for  the  epoch  1855. 

The  description  of  Argelander's  magnitude  scale  with  refer- 
ence to  modern  photometric  standards  will  be  given  later. 

The  southern  survey  of  the  heavens  has  been  extended  to 
declination  —42°  under  the  direction  of  Professor  Perrine  at 
the  Cordoba  Observatory  of  the  Argentine  Republic.  The 
charts  have  been  prepared  on  the  same  scale  as  the  northern 


50  THE  STUDY  OF  VARIABLE  STARS 

DM.  and  cover  the  degrees  in  declination  from  —22°  to  —42°, 
in  a  series  of  twelve  maps  which  have  already  been  distributed 
to  the  observatories.  It  is  to  be  hoped  that  the  survey  will  be 
completed  by  being  extended  to  the  South  Pole. 

A  very  valuable  series  of  charts  which  are  useful  to  workers 
on  faint  objects  are  those  prepared  by  Palisa  and  Wolf.  The 
purpose  of  these  charts  and  their  description  can  be  best  under- 
stood by  reading  the  statement  of  Palisa  in  the  Astrophysical 
Journal,  vol.  28,  p.  86 :  — 

Professor  Max  Wolf  of  Heidelberg  has  much  facilitated  my  task 
of  finding  and  observing  small  planets,  especially  those  of  the  faintest 
magnitudes,  by  sending  me  copies  of  his  photographs;  so  that  now 
it  takes  me  only  about  one-fourth  the  time  formerly  required  to  find 
them.  This  suggested  to  me  the  idea  that  it  would  be  a  great  advan- 
tage if  the  photographs  of  the  Heidelberg  Astrophysical  Institute  were 
made  available  for  every  observer  in  a  form  suitable  for  immediate 
use.  As  Professor  Wolf  had  intended  at  a  later  time  to  collect  his 
photographs  and  join  them  in  a  map,  he  kindly  offered  to  furnish 
positives  free  of  cost.  On  these  positives  a  reseau  is  then  carefully 
cut,  the  curvature  of  the  parallels  being  determined  by  the  stars 
themselves.  Each  plate  covers  fifty  square  degrees,  the  scale  being 
36  mm.  to  the  degree.  Contact  prints  are  then  made  from  the  posi- 
tives on  smooth  but  not  glossy  bromide  paper;  and  the  necessary 
text,  including  the  numbers  for  right  ascension  and  declination,  is 
then  printed  on  the  sheets,  which  admit  of  pencil  entries  and  erasures. 

I  have  not  attached  a  scale  of  magnitudes  to  these  maps  for  two 
reasons.  On  account  of  different  exposures,  disks  of  equal  size  do  not 
represent  the  same  magnitude  on  different  plates  and  even  on  a  single 
plate  the  scale  is  not  the  same  at  the  center  and  near  the  edge. 

Nine  volumes  of  this  work  have  been  published,  each  one 
containing  twenty  plates,  but  though  they  might  prove  very 
useful  for  observers  who  wish  to  identify  faint  stars,  their  cost 
is  quite  high  and  will  interfere  with  their  extensive  purchase 
except  by  observatories. 

There  are  a  few  miscellaneous  sets  of  charts  containing  faint 
stars,  but  they  are  usually  quite  limited  in  area  and  can  only 
be  used  for  special  purposes.  Among  them  may  be  mentioned 
the  Paris  Ecliptic  Charts,  prepared  by  the  Chacornac  and  the 


STAR  CHARTS  FOR  GENERAL  USE  51 

Henry  brothers;  those  made  at  Litchfield  Observatory  in 
Hamilton,  New  York,  by  C.  H.  F.  Peters,  and  those  of  the 
Carte  du  del.1 

The  close  of  this  chapter  on  star  charts  and  their  use  seems 
an  opportune  place  in  which  to  refer  briefly  to  precession  and 
give  some  directions  for  applying  it.  The  precession  of  the 
equinoxes  is  a  slipping  westward  of  the  equinox  along  the  eclip- 
tic at  a  rate  of  50".2  per  year.  It  therefore  changes  the  longi- 
tudes of  all  the  stars,  and  consequently  their  right  ascensions 
and  declinations.  When  a  catalogue  is  formed,  the  positions  of 
all  the  stars  in  it  must  be  referred  to  the  position  of  the  equinox 
for  a  certain  definite  time  which  is  called  the  epoch  of  the  cata- 
logue; for  example,  the  epocli  of  the  Durchmusterung  is  1855, 
while  the  actual  observations  extended  from  1852  to  1862.  In 
an  exact  catalogue  the  annual  precession  for  each  star  in  right 
ascension  and  declination  is  always  given,  but  in  some  of  the 
older  catalogues  this  has  been  omitted,  as  is  the  case  with  the 
Durchmusterung.  It  is  for  this  reason  that  the  precession  for 
ten  years  is  given  at  the  bottom  of  each  column.  It  may  be 
used  for  all  of  the  stars  in  its  column,  because  the  positions  are 
only  approximate.  The  sign  which  is  given  to  it  is  to  be  used  in 
carrying  the  star  forward  from  an  early  date  to  a  later  one, 
e.g.,  in  taking  it  from  the  catalogue  date,  1855  to  1900,  which  is 
the  date  of  the  Harvard  Variable  Star  Catalogue.  If  a  new 
variable  were  to  be  discovered  in  1915  and  its  position  deter- 
mined, and  we  wished  to  find  if  it  were  on  Argelander's  charts, 
it  would  be  necessary  to  apply  the  precession  for  sixty  years 
with  the  opposite  sign  in  order  to  carry  it  back  to  the  proper 
date.  It  is  for  this  reason  that  some  catalogues  of  variables 
give  the  star  positions  for  1855  in  order  to  facilitate  their  loca- 
tion on  the  Durchmusterung  charts. 

1  H.  H.  Turner,  The  Great  Star  Map  (E.  P.  Dutton  &  Co.,  1912),  con- 
tains a  full  account  of  the  formation  of  these  important  charts. 


CHAPTER  III 

STAR  CHARTS  FOR  VARIABLES 

THE  present  chapter  deals  with  charts  which  have  been  pub- 
lished especially  for  the  use  of  variable  star  observers.  The 
most  important  of  them  was  prepared  by  the  Reverend  J.  G. 
Hagen,  S.J.,  during  the  years  1899  to  1908,  the  work  being 
begun  at  the  Georgetown  College  Observatory,  Washington, 
and  finished  at  the  Vatican  Observatory,  of  which  he  is  now 
the  director.  It  would  be  impossible  to  overestimate  their 
value.  The  introduction,  unfortunately,  is  in  Latin,  and  many 
of  the  important  points  will  ordinarily  escape  the  reader,  and 
hence  it  seems  highly  desirable  to  give  a  full  and  rather  free 
rendering  of  the  original  text.  However,  on  account  of  their 
technical  character  the  following  pages,  beyond  the  opening 
paragraphs,  will  not  be  of  particular  interest  to  the  general 
reader,  being  addressed  rather  to  the  specialist  in  variables, 
who  has  before  him  the  charts  and  the  accompanying  catalogue 
sheets  of  this  remarkable  series. 

The  title  of  the  work  is  Atlas  Stellarum  Variabilium,  which 
is  abbreviated  by  Hagen  as  ASV.  It  is  in  six  series,  each  of 
which  consists  of  two  portfolios,  one  containing  the  charts,  and 
the  other  the  catalogue  sheets  for  the  comparison  stars.  The 
selection  of  the  stars  included  in  each  series  depends  upon  the 
extent  of  their  light  variation.  The  first  three  contain  stars 
whose  minimum  light  is  below  the  tenth  magnitude.  They  are 
further  divided  into  zones  according  to  the  declination.  Series 
I,  published  1899,  contains  stars  lying  between  the  declinations 
-25°  and  0°;  Series  II,  published  1899,0°  to +25°;  Series  III, 
published  1900, +25°  to +90°;  Series  IV,  published  1907,  con- 
tains those  variables  the  light  of  which  at  minimum  is  visible 
in  instruments  of  moderate  size,  and  for  which  both  declination 
and  magnitude  are  within  the  limits  of  the  DM.  charts;  Series 


STAR  CHARTS  FOR  VARIABLES  53 

V,  published  1906,  contains  variables  scattered  over  the  entire 
sky,  whose  minimum  light  is  greater  than  seventh  magnitude; 
Series  VI,  published  1908,  is  supplementary  to  Series  I,  II, 
and  III. 

The  preface,  which  is  the  same  for  Series  I,  II,  and  III,  con- 
tains the  following  descriptive  statements.  The  heading  of  the 
charts  contains  all  the  material  which  is  necessary  for  obser- 
vational use  at  night,  and  was  taken  largely  from  Chandler's 
Third  Catalogue,  after  everything  had  been  verified  from  obser- 
vations made  at  Georgetown.  At  the  top  is  given  the  Chandler 
number;  under  it  the  name  of  the  star.  The  next  line  contains 
the  right  ascension  and  declination  for  1900,  with  the  annual 
precession.  The  fourth  line  contains  at  the  left  the  color,  reck- 
oned on  the  scale  of  10,  0  representing  white,  and  10  represent- 
ing red.  Following  this  is  a  Roman  numeral,  which  indicates 
the  spectral  type  according  to  Secchi's  classification.  At  the 
right  are  the  magnitudes  at  maximum  and  minimum.  On  the 
lower  margin  of  the  chart  is  arranged  a  row  of  small  blackened 
circles,  indicating  the  magnitudes  of  the  stars,  and  below  these 
is  given  the  number  of  the  series.  If  the  region  of  the  variable 
is  found  on  other  charts,  such  as  the  Paris  Ecliptic  Charts,  and 
the  Clinton  Charts  of  Peters,  a  statement  to  this  effect  is  made. 

The  chart  itself,  which  is  beautifully  printed  on  heavy  paper, 
represents  a  region  of  the  sky  1°  square,  with  the  variable  at 
the  center,  and  is  divided  into  two  main  parts.  The  central 
square,  30'  on  each  side,  contains  nearly  all  the  stars  which  are 
easily  visible  with  the  Georgetown  telescope  and  whose  light  is 
as  faint  as  that  of  the  variable.  The  outer  part  contains  all  the 
stars  in  that  region  which  are  found  in  the  DM.  catalogue,  and 
some  additional  stars,  which  are  inserted  where  there  is  danger 
of  misidentification.  If  a  bright  star  is  in  the  neighborhood, 
but  a  little  too  far  away  to  appear  on  the  chart,  an  arrow  placed 
at  the  proper  declination  indicates  its  direction  in  right  ascen- 
sion. Heavy  lines  separate  the  inner  square  from  the  rest  of 
the  chart,  and  finer  lines  form  a  network  in  which  the  squares 
are  5'  wide.  On  the  margin  are  printed  numbers  which  indicate 


54  THE  STUDY  OF  VARIABLE  STARS 

the  distance  in  right  ascension  and  declination,  counted  from 
the  center.  In  the  latter  co-ordinate  they  are  5'  apart,  but  in 
right  ascension  the  difference  depends  upon  the  cosine  of  the 
declination  at  the  center.  For  example,  at  zero  degrees  they 
read:  Om,  20s,  40s,  lm,  but  at  sixty  degrees  they  read:  Om40s, 
Im209,  2m,  etc. 

Since  the  charts  are  made  for  use  with  the  telescope,  they  are 
inverted  in  direction;  north  is  at  the  bottom  of  the  figure,  east 
at  the  right,  south  at  the  top,  and  west  at  the  left.  The  eastern 
part  of  the  field  is  frequently  called  the  following  edge,  and  the 
western  the  preceding  edge,  since  that  is  the  order  in  which  the 
stars  move  by  diurnal  motion. 

The  variable  is  represented  on  the  chart  by  a  dot  with  a 
circle  around  it,  the  former  indicating  the  magnitude  at  mini- 
mum, and  the  latter  that  at  maximum. 

The  catalogue  sheets  containing  the  comparison  stars  are 
printed  on  paper  of  about  the  same  size  as  the  charts.  In  the 
upper  corner  of  each  sheet  is  the  number  of  the  series.  The 
material  in  the  headings  is  again  taken  from  Chandler's  Third 
Catalogue.  It  contains  the  Chandler  number,  the  name  of  the 
star,  the  position  for  1855,  and  the  elements,  that  is,  the  epoch 
and  the  period,  the  former  being  expressed  both  as  a  Julian 
Day  and  as  the  calendar  date.  The  contents  of  the  columns 
may  be  described  as  follows,  being  the  same  for  all  of  the  first 
three  series.  The  first  column  contains  the  current  numbers  of 
the  comparison  stars,  which  are  arranged  in  order  of  magnitude. 
The  second  column,  headed  "Gradus,"  contains  the  grades,  or 
steps;  the  third  column,  the  magnitudes  which  are  deduced 
from  the  grades;  and  the  fourth  column,  the  magnitudes  taken 
from  the  BD.,  when  the  star  occurs  in  that  catalogue.  The  next 
two  columns  give  the  quantities  Aa  and  AS,  or  the  differences 
in  right  ascension  and  declination,  counted  from  the  variable 
itself  and  referred  to  the  epoch  1900.  These  differences,  when 
added  to  the  position  of  the  variable  for  1855,  will  give  approxi- 
mately the  positions  of  the  comparison  stars  for  the  date  of  the 
Durchmusterung,  and  hence  will  aid  in  the  identification  of  the 


STAR  CHARTS  FOR  VARIABLES  55 

stars  in  the  catalogue.  The  positions  given  on  the  Hagen  Charts 
are  for  1900. 

The  brightness  of  the  stars  was  not  observed  in  such  a  way 
that  the  magnitudes  could  be  immediately  assigned  to  them, 
but  without  any  photometric  assistance  the  grade,  or  step,  was 
estimated  by  which  one  star  differed  from  another  a  little 
brighter  or  a  little  fainter.  In  this  way  the  brighter  stars  of  the 
three  series  were  compared,  with  a  small  instrument  of  4.8 
inches  aperture,  between  the  years  1892  and  1895,  and  again 
with  a  larger  instrument  between  the  years  1895  and  1898, 
at  which  time  the  fainter  stars  were  also  observed.  Thus  the 
grades  of  the  brighter  stars  were  found  by  four  independent 
determinations,  and  those  of  the  fainter  stars  by  two.  These 
partial  sequences  were  then  put  together  in  one  series,  and  the 
stars  arranged  in  order  of  grade  from  the  brightest  to  the  faint- 
est; and  they  are  so  placed  in  the  list.  The  method  of  convert- 
ing the  grades  into  magnitudes  is  of  importance,  since  it  is 
necessary  to  know  upon  what  standards  the  magnitudes  of  the 
comparison  stars  are  based  before  observations  of  the  variable 
can  be  combined  with  those  made  with  other  comparison  stars, 
Hagen  makes  it  quite  clear  that  he  considers  his  magnitudes 
only  relative,  and  not  absolute,  but  he  still  believes  that  they 
serve  the  purpose  for  which  they  were  intended.  At  the  time 
when  his  charts  for  the  first  three  series  were  issued,  there  were 
no  photometric  observations  of  faint  stars,  and  not  many  for 
stars  of  the  seventh  and  eighth  magnitudes.  The  only  standard 
which  had  any  degree  of  uniformity  was  found  in  the  magni- 
tudes of  the  Durchmusterung.  Therefore,  he  connected  his 
grades  with  the  magnitudes  of  the  BD.  stars  which  were  found 
on  his  maps,  so  that  the  two  scales  fitted  together  between  the 
magnitudes  7.0  and  10.0. 

He  derived  by  this  means  a  formula  which  gave  expression 
to  the  relation.  For  example,  star  no.  1,  Series  I,  is  S  Ceti;  his 
formula  for  converting  the  grades  into  magnitudes  is, 

M  =  8.9  +  0.071  (G  -  17.8), 
in  which  G  stands  for  the  grade  given  in  the  second  column, 


56  THE  STUDY  OF  VARIABLE  STARS 

0.071  is  the  value  of  one  grade  expressed  in  magnitudes,  and 
8.9  is  the  magnitude  for  a  star  of  grade  17.8.  The  use  of  this 
formula  is  continued  for  the  very  faintest  stars.  Each  chart 
thus  has  a  formula  of  its  own,  and  it  is  only  through  the  com- 
mon use  of  the  BD.  magnitudes  that  a  uniformity  of  scale 
exists  between  the  different  charts.  For  stars  below  10.0  ing., 
which  is  the  lower  limit  of  the  BD.,  even  this  is  not  attained, 
for  as  the  scale  is  extended  downward,  the  lower  limit  of  magni- 
tude will  not  be  the  same  for  all  the  charts,  for  several  reasons. 
Firstly,  the  conditions  under  which  it  was  obtained  will  not  be 
the  same,  owing  to  the  difference  in  altitude  and  atmospheric 
conditions,  with  the  result  that  the  lower  limit  visible  with  the 
Georgetown  telescope  of  twelve  inches  aperture  will  vary  from 
11.5  to  13.5  mg.  The  light  ratios,  in  passing  from  one  magni- 
tude to  the  next,  cannot  be  assumed  constant  without  the  use 
of  a  photometer.  Furthermore,  the  Durchmusterung  scale  is 
probably  not  uniform  all  over  the  sky,  and  if  the  reference  stars 
of  9.5  mg.  or  10.0  mg.  are  too  bright  or  too  faint,  the  resulting 
magnitudes  of  the  fainter  stars  will  be  similarly  affected.  The 
relative  brightness,  however,  as  indicated  by  the  steps,  will 
remain  unchanged.  The  magnitudes  were  intended  primarily 
for  representation  on  the  charts  for  purposes  of  identification; 
the  observer  need  not  use  them  in  his  observations  or  computa- 
tions. He  need  not  even  use  the  step  values  if  he  prefers  those 
derived  from  his  own  comparisons,  but  they  may  be  of  service 
at  any  time  when  photometric  observations  of  a  few  comparison 
stars  have  been  made,  in  order  to  find  the  relation  between  the 
photometric  scale  and  the  visual  scale.  They  may  thus  be 
adapted  to  any  scale.  The  process  of  adapting  the  Hagen 
grades  for  the  first  three  charts  to  the  Harvard  photometric 
scale  will  be  found  in  Annals,  H.C.O.,  vol.  37,  part  n. 

The  positions  of  the  stars  for  the  charts  were  determined 
with  the  aid  of  a  semicircular  glass  disk  in  the  eyepiece,  so 
inserted  that  the  diameter  served  as  an  hour-circle.  On  the 
glass  were  drawn  perpendicular  lines,  so  heavy  that  they  could 
be  perceived  by  the  natural  light  of  the  sky.  The  scale  was 


STAR  CHARTS  FOR  VARIABLES  57 

divided  into  ten  parts  of  3'  each.  It  will  be  remembered  that 
this  method  was  used  by  Argelander.  The  declinations  were 
estimated  to  the  tenth  part  of  an  interval,  or  0'.3,  without 
haste,  while  the  telescope  was  being  carried  by  the  driving 
clock,  from  which  method  an  error  of  0'.3  or  perhaps  0'.6  can 
be  expected  in  the  declination.  The  right  ascensions  were 
determined  by  three  observations  on  the  chronograph,  and 
probably,  except  in  the  case  of  the  fainter  stars,  do  not  vary 
more  than  a  second  from  the  correct  value.  The  amount  by 
which  the  scale  was  inclined  to  the  hour-circle  was  determined 
for  the  individual  charts  from  many  stars  whose  positions  were 
taken  from  different  catalogues  already  published,  or  from 
stars  in  the  A.G.C.,  extracts  from  which  were  sent  to  Hagen 
before  being  set  up  in  type,  and  also  from  observations  made 
with  the  meridian  instrument  at  the  Georgetown  Observatory. 
The  epoch  to  which  the  quantities  Aa  and  AS  are  referred  is 
the  year  1900.  It  is  to  be  noted  that  the  positions  of  the  stars 
outside  the  limits  of  the  chart  are  taken  for  the  most  part  from 
the  ED. 

The  last  column  contains  the  notes,  for  which  little  explana- 
tion is  needed.  "  Duplices  "  indicates  stars  the  component  parts 
of  which  cannot  easily  be  separated.  The  numbers  which  are 
added  from  the  various  catalogues  of  variable  stars  require  no 
particular  explanation.  Another  sort  of  note  is  the  abbrevia- 
tion, either  Sch.  or  Ch.9  by  which  it  is  indicated  that  the  stars 
so  designated  are  near  the  variable  in  the  catalogue  either  of 
Schonfeld  or  Chandler.  Since  the  brightness  of  some  of  the 
comparison  stars  exceeds  7.0  mg.,  the  above  mentioned  method 
could  not  be  adopted  for  determining  their  magnitudes,  and 
hence  these  are  added  in  the  notes,  being  taken  from  other 
sources,  which  depend  upon  the  declination  of  the  star;  e.g., 
the  Cordoba  Durchmusterung  (CD.),  or  the  Potsdam  Durch- 
musterung  (PD.). 

In  the  remaining  lines  of  the  Introduction  Father  Hagen 
states  that  he  himself  was  responsible  for  the  observations  of 
the  positions  and  grades  of  the  stars,  and  that  the  computations 


58  THE  STUDY  OF  VARIABLE  STARS 

of  the  magnitudes  of  the  stars  and  the  inclinations  of  the  scale 
were  made  by  his  associates.  He  also  expresses  his  thanks  to 
those  who  had  assisted  him  in  collecting  the  material  for  the 
charts. 

The  fourth  series  was  published  in  1907,  after  the  appearance 
of  volume  37  of  the  Annals,  above  referred  to,  which  contained 
the  discussion  of  the  relation  between  the  Harvard  photometric 
magnitudes  and  the  Hagen  grades  for  many  stars  of  the  first 
three  series;  and  hence  Hagen  had  the  opportunity  to  profit  by 
further  co-operation  with  the  Harvard  Observatory,  as  will  be 
described  later  on,  in  connection  with  the  magnitudes  of  the 
comparison  stars.  This  fourth  series  was  prepared  for  the 
observation  of  those  stars  whose  minimum  light  could  be 
observed  by  instruments  of  small  aperture,  that  is,  of  from 
three  to  six  inches.  Therefore  the  limiting  magnitude  of  the 
stars  delineated  upon  the  charts  is  almost  the  same  as  that  of 
the  Durchmusterung  catalogue.  The  headings  of  the  charts  are 
practically  the  same  as  for  the  first  three  series,  the  values, 
however,  being  taken  from  the  latest  sources.  The  only  differ- 
ence between  the  charts  of  this  series  and  those  of  the  first 
three  is  in  the  scale  used;  the  outside  square  is  now  2°  wide, 
and  the  small  ones  10'.  A  further  reference  to  this  fact  will  be 
found  in  Chapter  XV.  The  ratio  of  star  density  between  the 
inner  and  outer  regions  of  the  chart  is  the  same  as  in  the  earlier 
series.  Not  only  are  all  the  stars  of  the  DM.  included,  but 
fainter  stars  are  added  wherever  they  are  useful  for  observing 
the  minimum  light  of  the  variable  or  for  making  the  configura- 
tion more  certain.  In  the  area  surrounding  the  inner  square 
the  lower  limit  of  magnitude  is  between  8.0  mg.  and  9.0  mg. 
whenever  this  seems  desirable.  Those  charts  which  contain  the 
variables  in  Chandler's  Third  Catalogue  were  drawn  by  J. 
Hisgen  at  the  Georgetown  Observatory  and  later  compared 
with  the  sky  at  Valkenburg,  Netherlands. 

On  the  catalogue  sheets  the  headings  are  practically  the  same 
as  for  the  first  three  series,  the  sources  being  different  in  some 
cases.  In  the  case  of  the  Algol  type  the  periods  only  are  given, 


STAR  CHARTS  FOR  VARIABLES 


59 


since  the  times  of  minimum  light  can  be  taken  more  accurately 
and  quickly  from  the  Ephemerides.  The  material  found  on  the 
sheets,  however,  is  considerably  different.  The  first  column 
contains  the  current  number,  the  second  and  third  the  BD. 
number  and  magnitude.  The  fourth  column,  which  is  headed 
HP.  (Harvard  Photometry),  contains  for  several  of  the  stars 
photometric  magnitudes  which  were  communicated  to  Hagen 
directly  by  Pickering  from  observations  made  just  previously 
at  the  Harvard  College  Observatory.  How  these  were  used  in 
determining  the  magnitudes  will  be  described  presently.  The 
column  headed  "Gradus"  contains  sometimes  two  sets  of  num- 
bers, one  of  which  was  derived  from  observations  made  by 
Hagen,  and  the  other  from  those  by  Hisgen.  Therefore,  if  the 
two  sets  are  present,  the  first  are  due  to  Hagen,  and  the 
others  to  Hisgen;  if  only  one,  it  is  the  work  of  Hagen  alone. 

The  magnitudes  were  obtained  by  the  same  general  method 
that  was  employed  in  the  preceding  series,  that  is,  by  connect- 
ing the  observed  grades  with  the  magnitudes  of  the  stars  which 
were  already  known.  However,  for  this  series  they  rest  upon 


Mr 

6.0 

( 

TO 
8.0 
9.0 
10-0 
1  1.0 

Gr 

> 

b& 

CT- 

o 

^k 

^ 

^ 

V 

^ 

-*^0 

-< 

^ 

I 

^ 

0       10       2.0      30      40      £0      60       70      80      90      100     110 

Figure  16 

MAGNITUDE  CURVE  FOR  RV  HYDRAE 


60 


THE  STUDY  OF  VARIABLE  STARS 
TABLE  I 


Star  No. 

H.P. 

Or. 

Mag. 

Star  No. 

H.P. 

Gr. 

Mag. 

1 

6.82 

0 

6.7 

23 

9.37 

62 

9.4 

2 

6.48 

0 

6.7 

24 

63 

9.4 

3 

6.95 

4 

6.9 

25 

66 

9.5 

4 

7.18 

6 

7.0 

26 

71 

9.8 

5 

7.52 

11 

7.2 

27 

73 

9.8 

6 

7.86 

17 

7.5 

28 

9.77 

77 

10.0 

7 

7.68 

22 

7.7 

29 

78 

10.0 

8 

25 

7.9 

30 

84 

10.2 

9 

27 

8.0 

31 

10.08 

85 

10.3 

10 

27 

8.0 

32 

88 

10.4 

11 

8.11 

31 

8.1 

33 

90 

10.5 

12 

8.27 

33 

8.2 

34 

91 

10.5 

13 

36 

8.3 

35 

93 

10.6 

14 

38 

8.4 

36 

10.57 

97 

10.7 

15 

43 

8.6 

37 

97 

10.7 

16 

8.65 

47 

8.8 

38 

100 

10.8 

17 

51 

8.9 

39 

10.70 

101 

10.8 

18 

55 

9.1 

40 

101 

10.8 

19 

57 

9.2 

41 

10.56 

101 

10.8 

20 

9.26 

57 

9.2 

42 

103 

10.9 

21 

59 

9.2 

43 

106 

11.0 

22 

60 

9.3 

44 

11.33 

111 

11.2 

a  much  better  foundation,  namely,  upon  the  photometric  mag- 
nitudes communicated  by  Pickering  instead  of  on  the  Durch- 
musterung  values.  The  process  is  described  as  follows.  Points 
were  plotted  using  the  magnitudes  as  ordinates,  and  the  Hagen 
grades  as  abscissas,  and  curves  drawn  through  them  which 
defined  the  magnitude  for  each  grade.  These  curves  differed 
for  the  different  charts,  but  did  not  deviate  much  from  the 
straight  line.  Where  there  were  two  sets  of  grades  the  magni- 
tudes were  determined  from  both  curves,  and  the  arithmetical 
mean  of  the  results  taken.  Their  disagreement  is  for  the  most 


STAR  CHARTS  FOR  VARIABLES  61 

part  between  the  limits  i  0.1  mg.  and  i  0.2  mg.,  although 
when  the  curve  is  extended  beyond  the  limits  of  the  stars  fur- 
nished by  Pickering,  it  occasionally  increases  to  larger  values. 
The  magnitudes  thus  determined  appear  in  column  six.  Above 
are  given  in  a  table  the  necessary  values  for  constructing  the 
magnitude  curve  for  RV  Hydrae.  The  points  on  the  curve 
were  plotted  by  using  the  Hagen  grades  as  abscissas  and  the 
Harvard  magnitudes  (HP.)  as  ordinates.  The  magnitudes  in 
the  fourth  column  were  read  from  the  curve. 

The  values  for  Aa  and  AS  were  computed  for  the  year  1900, 
though  the  positions  of  the  variables  in  the  headings  of  the 
catalogue  sheets  are  for  1855.  The  positions  of  the  brighter 
stars  were  taken  from  the  catalogues  of  the  A.G.C.  For  the 
fainter  stars  they  were  obtained  partly  from  observations  made 
at  Georgetown,  partly  from  measurements  made  at  Harvard, 
on  photometric  plates,  and  partly  from  the  ED. 

In  the  column  "Notae"  may  be  found  magnitudes  and  colors 
taken  from  the  Potsdam  Durchmusterung,  the  letters  of  Bayer, 
and  the  numbers  of  Flamsteed.  The  abbreviation  "dpi."  refers 
to  those  stars  the  components  of  which  cannot  be  observed 
separately,  and  which  hence  should  not  be  used  as  comparison 
stars  in  measuring  the  light  from  the  variable.  Furthermore, 
if  the  "dpi."  precedes  the  name  of  a  catalogue,  it  indicates  that 
the  observation  was  first  made  with  the  telescope  at  George- 
town or  Valkenburg,  but  if  it  follows,  then  it  was  taken  directly 
from  the  catalogue;  e.g.,  RT  Hydrae,  star  23,  "dpi.  A.G.C.  9.5 
prec.,"  and  X  Monocerotis,  star  17,  "A.G.C.  dpi.  9.0."  The 
small  letters  enclosed  in  parentheses  designate  the  colors  as 
assigned  by  Hisgen,  and  signify  the  same  colors  as  the  corre- 
sponding capital  letters  from  the  PD.  catalogue;  e.g.,  "(r)" 
denotes  the  same  color  as  "R." 

The  fifth  series  contains  stars  which  are  visible  to  the  naked 
eye.  A  few  are  inserted  from  the  other  series  also  if  a  large  part 
of  the  variation  is  visible  to  the  naked  eye  or  can  be  observed 
with  hand  instruments.  These  are  x  Cygni,  o  Ceti,  R  Hydrae, 
R  Carinae,  and  in  addition  f]  Carinae.  There  are  other  stars, 


62  THE  STUDY  OF  VARIABLE  STARS 

twenty  six  in  number,  not  properly  belonging  to  this  series, 
whose  minimum  light  is  below  seventh  magnitude,  which  are 
included  on  the  charts  because  they  can  be  recognized  and 
observed  at  maximum,  being  then  lucid. 

In  making  the  charts  Hagen  received  the  assistance  of  vari- 
ous helpers  at  the  Georgetown  Observatory,  each  chart  being 
carefully  compared  with  the  sky.  Fainter  stars  were  added 
when  necessary  for  observing  the  light  at  minimum  or  for 
avoiding  ambiguity.  The  last  four  charts,  which  give  the  region 
about  the  south  pole,  were  drawn  by  Goetz  at  Bulawayo, 
Rhodesia.  The  charts,  while  made  on  the  same  general  plan 
as  those  in  the  earlier  series,  differ  somewhat  in  scale,  each  one 
being  suited  to  the  particular  variable.  The  scale  of  projection 
and  the  co-ordinates  of  the  center  of  each  will  be  found  in 
Table  I  of  the  preface.  In  the  region  immediately  surrounding 
those  variables  which  are  faint  at  minimum,  the  stars  are 
denser  than  in  the  remaining  portion,  which  does  not  contain 
any  fainter  than  the  fifth  magnitude.  The  catalogue  sheets 
which  accompany  the  charts  are  considerably  different  from 
those  of  the  previous  series.  The  heading  gives  the  Chandler 
number,  the  name  of  the  star,  and  the  character  and  elements 
of  the  variation  taken  from  Chandler's  Third  Catalogue.  In 
the  case  of  stars  of  the  Algol  type,  variations  in  the  length  of 
the  period  are  only  indicated,  since  it  is  much  easier  and  more 
accurate  to  take  the  times  of  minimum  directly  from  ephemeri- 
des  especially  prepared  for  the  purpose  than  to  compute  them 
from  the  elements.  They  are  now  given  in  full  in  Hartwig's 
yearly  catalogue. 

Not  only  are  the  stars  named  by  their  constellations  (col- 
umn one),  according  to  the  divisions  made  by  Argelander,  Heis, 
and  Gould,  but  also  there  are  added  the  Bayer  letters,  the 
Flamsteed  numbers,  or  those  from  the  Uranometria  Argentina 
in  the  case  of  far  southern  stars.  If  the  stars  are  placed  in  other 
constellations  by  other  authorities,  it  is  so  stated  in  the  notes. 
Following  these  designations  in  the  fifth  column  are  numbers 
taken  from  the  BD.  or  Cordoba  catalogue,  or  from  the  cata- 


STAR  CHARTS  FOR  VARIABLES  63 

logue  in  Annals,  H.C.O.,  vol.  34,  Southern  Meridian  Photometry. 
The  positions  of  the  stars  are  given  in  the  order  of  their  right 
ascensions  in  the  different  constellations.  Most  of  them  have 
been  taken  from  the  best  catalogues,  such  as  the  Berliner  Jahr- 
buch,  A.G.C.  Zones,  Cordoba  General  Catalogue,  a  few  from  the 
BD.  or  CD.,  and  all  have  been  reduced  to  the  year  1900. 

The  magnitudes  of  the  stars  have  been  taken  from  the 
Potsdam  Photometric  Durchmusterung  (PD.),  the  Harvard 
Photometry  (HP.),  or  the  Uranometria  Argentina  (UA.).  In 
the  last  four,  in  the  place  of  PD.  is  given  LM .,  by  which  is 
signified  the  magnitudes  determined  by  Dr.  Roberts,  Love- 
dale,1  South  Africa.  In  the  last  column,  headed  "Notae,"  may 
be  found  the  variation  in  brightness  of  the  variable,  taken  from 
Chandler's  Third  Catalogue  and  printed  in  bold  type,  differences 
in  the  designations  of  stars,  the  numbers  of  clusters  and  nebu- 
lae taken  from  other  catalogues,  colors,  etc.  Regarding  color, 
those  numbers  which  immediately  follow  the  letters  Kr.  refer 
to  the  work  of  Krueger  entitled  Catalogue  of  Colored  Stars, 
[e.g.,  6?6  Kr.  1282]  or  if  a  number  be  enclosed  in  parentheses, 
it  is  taken  from  the  supplement.  If  the  letters  precede,  they 
signify  that  the  colors  of  the  stars  were  measured  by  the  same 
authority,  but  in  accordance  with  the  scale  of  Schmidt,  and  not 
that  of  Chandler,  as  in  the  remaining  series.  As  the  purpose  has 
been  only  to  call  attention  to  stars  which  are  too  highly  colored 
to  be  used  for  comparison  stars,  numbers  lower  on  the  scale 
than  4.0  have  not  been  included,  since  they  represent  colors 
which  do  not  differ  much  from  white.  No  effort  was  made 
toward  a  critical  comparison  of  this  catalogue  with  others,  as 
PD.,  or  Osthoff.  It  is  enough  to  state  that  the  scales  of  Osthoff 

1  Note  from  Dr.  Roberts,  explaining  his  method  of  estimating  magnitudes, 
taken  from  a  letter.  By  experiment  he  had  determined  three  limiting  magni- 
tudes to  which  he  referred  the  other  magnitudes  by  means  of  grades.  He  as- 
signed mg.  6.8  to  the  faintest  star  which  could  with  difficulty  be  seen  with  the 
naked  eye,  9.2  mg.  to  stars  which  could  with  difficulty  be  seen  with  a  one- 
inch  telescope,  and  11.4  mg.  to  those  which  were  just  seen  with  a  three  inch. 
He  arranged  the  stars  in  sequences  of  grades  between  any  two  of  these  limit- 
ing magnitudes  in  opposite  directions,  starting  sometimes  from  the  upper 
limit  and  sometimes  from  the  lower. 


64  THE  STUDY  OF  VARIABLE  STARS 

and  Krueger  (with  the  general  exception  of  O-K  =  +1?3) 
agree  with  Schmidt,  and  that  of  the  PD.  very  nearly  with 
Chandler's. 

In  the  last  four  folios  are  inserted  notes  from  the  UA.  in 
which  the  letter  "r"  indicates  red  stars,  and  "c"  stands  for 
other  colors.  Other  notes  are  self-explanatory.  At  the  end  of 
each  is  added  a  table  giving  a  list  of  the  comparison  stars  by 
number  which  have  specially  been  used  by  observers  accus- 
tomed to  work  on  these  variables,  from  which  it  may  easily  be 
seen  which  are  the  most  suitable  to  use.  Table  II  of  the  preface 
contains  a  list  of  the  abbreviations  of  the  authorities. 

Special  reference  was  made  to  an  instrument  suitable  for 
observing  the  stars  in  this  series.  In  fact  the  division  of  the 
stars  into  five  groups  had  actually  been  made  in  order  that 
those  in  each  group  might  be  observed  with  the  same  sort  of 
instrument.  The  first  three  contain  stars  suited  for  larger  tele- 
scopes; the  fourth,  since  it  contains  stars  within  the  limits  of 
thel?Z).,is  especially  adapted  to  smaller  instruments;  and  the 
fifth  is  for  observation  with  the  naked  eye,  or  small  instruments 
which  may  be  held  in  the  hand.  Hagen  then  describes  the 
Steinheil  binoculars,  which  have  an  aperture  of  34  mm.  The 
image  is  enlarged  five  diameters,  and  the  intensity  of  its  light 
is  increased  forty-nine  times. 

The  sixth  series  of  the  charts  is  intended  to  be  supplementary 
to  the  first  three,  and  is  prepared  in  exactly  the  same  manner, 
except  that  in  the  catalogue  sheets  an  additional  column,  the 
fifth,  occurs,  which  contains  the  magnitudes  according  to  the 
scale  of  the  Harvard  Photometry.  These  were  deduced  from 
certain  stars  among  them,  the  magnitudes  of  which  were  fur- 
nished as  standards  by  Professor  Pickering.  They  are  indicated 
in  this  column  by  having  the  magnitudes  printed  in  bold  type. 
In  using  them  their  magnitudes  were  plotted  as  ordinates  with 
the  Hagen  grades  as  abscissas;  curves  were  then  drawn  through 
the  points  from  which  the  magnitudes  were  read  for  all  the 
stars,  as  explained  for  Series  IV. 

There  were  two  reasons  why  the  column  HP.  was  added;  the 


STAR  CHARTS  FOR  VARIABLES  65 

first  that  the  relation  which  existed  between  the  magnitudes 
of  Series  I,  II,  and  III  and  the  HP.  system  might  appear  more 
clearly,  and  the  second  that  certain  tables  in  the  Annals, 
H.C.O.,  vol.  37,  which  were  prepared  for  the  purpose  of  con- 
verting the  Hagen  grades  into  Harvard  magnitudes,  might  be 
extended  to  the  charts  in  Series  VI.  This  column  is  especially 
important  because  it  thus  exhibits  the  relation  between  the 
magnitudes  derived  from  the  two  systems.  The  maximum  or 
minimum  light  in  the  separate  folios  indicated  in  the  headings 
is  taken  from  the  elements  contained  in  the  Third  Catalogue  of 
Chandler  and  its  revision,  or  from  Pickering's  second  cata- 
logue,1 or  from  a  new  catalogue,  the  material  for  which  is  being 
prepared  for  the  Committee  of  the  Astronomische  Gesellschaft, 
and  was  communicated  in  advance  by  Dr.  Miiller. 

In  addition  to  the  introductions  for  each  series  which  have 
just  been  described,  there  is  also  a  General  Index  giving  the 
number  of  each  star  in  its  series,  the  arrangement  being 
according  to  the  order  of  right  ascension.  A  second  table  is  an 
Index  to  the  Constellations,  in  which  the  stars  are  arranged 
according  to  the  constellations.  A  third  table  furnishes  a  key 
connecting  the  present  system  of  nomenclature  with  that  for- 
merly in  use  by  the  Bureau  des  Longitudes,  but  now  superseded. 

We  shall  now  describe  the  maps  which  have  been  especially 
prepared  for  the  use  of  variable  star  observers  at  the  Harvard 
College  Observatory,  but  in  the  present  chapter  only  the 
method  of  preparing  the  maps  will  be  described,  leaving  the 
method  of  determining  the  magnitudes  to  be  given  later.  The 
Durchmusterung  maps  have  been  made  practically  useful  for 
the  observer  by  selecting  a  region  3°  square  surrounding  the 
variable  and  enlarging  it  photographically.  On  the  negative 
before  printing  are  written  the  letters  of  the  comparison  stars. 
These  are  then  printed  on  heavy  paper  8  x  10  inches  in  size, 
which  can  be  used  with  the  telescope  very  conveniently.  They 
are  especially  suitable  for  a  small  instrument  and  for  variables 
which  do  not  go  below  9th  or  10th  magnitude  at  minimum, 
i  Annals,  H.C.O.,  65. 


66  THE  STUDY  OF  VARIABLE  STARS 

The  correct  magnitudes  of  the  comparison  stars  accompany 
the  maps.  They  are  obtained  from  photometric  observations 
made  at  the  Harvard  Observatory  for  this  especial  purpose. 
More  recently,  in  preparing  these  enlargements  of  the  BD. 
charts,  instead  of  attaching  the  letters  to  the  comparison  stars 
the  magnitudes  have  been  used,  so  that  an  observer  can  obtain 
the  magnitude  of  the  variable  directly. 

The  Harvard  Observatory  has  also  prepared  for  its  use  other 
photographs  taken  directly  from  the  sky  with  exposures  of 
different  lengths,  the  variable  occupying  the  center  of  the  map. 
Sometimes  these  are  enlargements  of  negatives  already  ob- 
tained for  other  purposes.  On  these  charts  the  comparison 
stars  are  marked  either  with  magnitudes  or  letters.  The  magni- 
tudes are  used  for  all  the  brighter  stars  and  frequently  for  faint 
stars  down  to  magnitude  13.5.  Sometimes,  however,  the  num- 
bers extend  only  to  magnitude  13.0  and  letters  are  attached  to 
the  fainter  stars.  The  Director  of  the  Harvard  Observatory, 
Professor  E.  C.  Pickering,  whose  long-continued  interest  in 
variable  stars  is  well  known  to  the  astronomical  world,  has  at 
different  times  invited  the  co-operation  of  astronomers  and 
amateur  observers  in  the  study  of  variable  stars,  and  has  offered 
to  provide  photographic  maps  for  any  one  who  wishes  to  make 
use  of  them.  The  lists  of  the  comparison  stars  for  many  of  the 
variables  which  he  recommends  for  study  have  already  been 
published  in  Annals,  H.C.O.,  vols.  37  and  57,  and  others  will 
appear  later.  There  is  some  difference  of  opinion  in  regard  to 
the  systems  of  magnitudes  employed,  especially  in  the  case  of 
the  faint  stars,  but  the  observer  who  is  using  the  Harvard  maps 
is  recommended  to  adopt  those  given  on  the  maps,  since  they 
will  then  be  on  a  uniform  scale.  If  at  some  later  time  it  is 
thought  desirable  to  adopt  another  system,  the  change  can 
easily  be  made. 

For  observers  who  are  working  with  faint  stars  the  photo- 
graphs of  Parkhurst  will  be  of  very  great  use.  They  were  taken 
at  the  Yerkes  Observatory  with  a  twenty-four-inch  reflector, 
at  Father  Hagen's  request,  and  include  all  of  his  charts  in 


STAR  CHARTS  FOR  VARIABLES  67 

Series  I,  II,  III,  and  VI,  in  which  the  variable  reaches  the  13th 
magnitude  or  fainter  at  minimum.  Of  the  193  fields  in  the  four 
series  the  variable  falls  to  the  13th  magnitude  in  140  cases. 
These  140  fields  have  been  photographed,  and  negative  prints 
on  bromide  paper  8  x  10  inches  in  size  can  be  obtained  at  the 
Yerkes  Observatory.  The  scale  of  the  prints  is  10"  to  one  mm. 
Therefore  the  field  covered  will  be  0.8  of  a  degree  square.  For 
galactic  fields  crowded  with  stars,  prints  of  double  this  scale  can 
be  supplied.  The  name  of  the  field,  the  place  for  1900,  and  the 
orientation  are  to  be  marked  on  the  print.  The  variable  itself 
will  be  enclosed  in  a  small  circle.  The  Hagen  chart  can  serve  as 
an  index  to  these  photographs,  the  brighter  stars  be  identified 
on  his  lists,  and  with  the  known  scale  the  positions  of  the 
fainter  stars  relative  to  the  variable  can  be  determined.  The 
plates  have  been  taken  with  an  exposure  of  one  hour  and  show 
stars  to  the  16th  magnitude.1 

Another  set  of  maps  containing  the  faint  stars  surrounding 
twelve  variables  may  be  found  in  a  volume  by  Parkhurst  en- 
titled Researches  in  Stellar  Photometry.  This  work  was  carried 
on  largely  at  the  Yerkes  Observatory  and  many  of  the  com- 
parison stars  are  fainter  than  the  14th  magnitude.  The  maps 
and  magnitudes  will  be  especially  useful  to  those  who  are 
observing  these  particular  stars. 

Maps  for  individual  variables  are  scattered  through  vari- 
ous numbers  of  the  Astronomische  Nachrichten,  Astrophysical 
Journal,  etc.,  and  may  also  be  found  in  the  publications  of 
many  observatories. 

i  Ap.  J.,  28, 87. 


CHAPTER  IV 

CATALOGUES  OF  VARIABLES 

THERE  are  several  systems  for  naming  variable  stars,  but 
the  one  described  here  is  that  most  generally  in  use.  As  soon  as 
a  variable  star  is  discovered,  the  fact  is  communicated  to  the 
Astronomische  Nachrichten  for  publication.  The  editor  of  the 
journal  refers  it  to  a  committee  of  the  Astronomische  Gesell- 
schaft  which  has  the  matter  in  charge,  and  they  assign  to  the 
star  a  provisional  number  which  indicates  the  year  and  the 
order  of  discovery  in  the  year.  This  is  followed  by  the  name 
of  the  constellation;  e.g.,  89,  1914  Persei,  signifies  the  89th 
variable  discovered  during  the  year  1914  which  is  in  the  con- 
stellation of  Perseus. 

This  provisional  name  is  retained  until  the  variation  is  con- 
firmed and  the  elements  are  more  or  less  known.  A  permanent 
letter  is  then  assigned  to  it  in  accordance  with  the  following 
rule,  which  originated  with  Argelander.  The  first  variable  dis- 
covered in  a  constellation  is  given  the  letter  R,  the  second  S, 
and  so  on,  the  ninth  one  having  the  letter  Z.  The  tenth  star 
has  the  double  letter,  RR,  the  next  one  RS,  and  so  on  as  far  as 
RZ.  They  begin  again  with  SS,  and  continue  in  this  manner 
until  the  combination  ZZ  is  reached,  allowing  thus  for  fifty- 
four  variables  in  one  constellation.  It  became  evident  several 
years  ago  that  this  method  would  not  suffice,  and  some  other 
device  was  necessary.  The  committee  appointed  by  the 
Astronomische  Gesellschaft  reported  that  as  it  would  be  quite 
inconvenient  to  triple  the  letters,  the  best  plan  would  be  to 
return  to  the  first  letters  of  the  alphabet  and  beginning  with 

the  combination  AA,  AB, AZ,  BB, BZ,  continue 

as  far  as  QZ,  thus  adding  280  more  combinations.  This  was 
adopted,  and  the  first  constellation  to  which  it  was  applied  was 
Cygnus,  which  in  Hartwig's  ephemeris  for  1914  has  the  combi- 


CATALOGUES  OF  VARIABLES  69 

nation  BB,  while  Scorpio  has  AL,  and  Sagittarius,  AR.  This 
report  of  the  Astronomische  Gesellschaft  committee  may  be 
found  in  Astronomische  Nachrichten  4212.  At  frequent  inter- 
vals lists  of  variables  containing  the  provisional  and  perma- 
nent names  are  published  by  the  committee,  together  with 
other  important  information  and  notes,  e.g.,  Astronomische 
Nachrichten  4457. 

Another  method  of  designating  variables  by  numbers  was 
devised  at  the  Harvard  Observatory  and  is  in  general  use  there. 
Each  number  consists  of  six  figures,  the  first  two  of  which  give 
the  hours  of  right  ascension,  the  second  two  the  minutes,  and 
the  last  two  the  degrees  of  declination;  e.g.,  123961  is  the  Har- 
vard number  for  S  Ursae  Majoris,  and  shows  that  its  right 
ascension  is  12  hours  and  39  minutes,  and  its  declination  is  61°. 
If  the  star  is  south  of  the  equator  the  number  is  printed  in 
italics.  The  advantage  of  this  method  of  numbering  is  that  it 
is  easy  to  locate  the  variable  in  the  sky.  At  the  Harvard 
Observatory  members  of  the  staff  who  are  accustomed  to  using 
the  numbers  can  remember  them  and  set  without  looking  at 
the  map.  This,  however,  can  be  done  only  with  a  small  tele- 
scope, for  with  one  of  large  aperture  it  is  necessary  to  set  to  a 
tenth  of  a  degree  in  declination  and  hence  the  Harvard  number 
is  not  sufficient. 

Another  method  of  numbering  was  introduced  by  S.  C. 
Chandler  and  the  numbers  in  the  system  are  called  Chandler 
numbers.  They  are  obtained  by  reducing  the  right  ascension 
for  1900  to  seconds  and  dividing  by  ten;  e.g.,  the  Chandler 
number  for  S  Ursae  Majoris  is  4557,  which  is  obtained  from  the 
exact  right  ascension,  which  is  12h  39m  34s,  or  45,574  seconds. 
This  method,  which  was  extensively  used  some  years  ago,  is  no 
longer  employed. 

A  fourth  method  of  designation,  which  was  suggested  by 
Andre  in  his  Traite  d'Astronomie  Stellaire,  is  very  logical  but 
has  not  been  adopted  generally,  although  by  it  an  indefinite 
number  of  variables  in  a  constellation  can  be  included.  \The 
letter  V  followed  by  a  number  is  prefixed  to  the  name  of  the 


70  THE  STUDY  OF  VARIABLE  STARS 

constellation,  the  number  indicating  the  order  of  discovery  in 
that  constellation;  e.g.,  V  8  Draconis  would  signify  the  eighth 
variable  discovered  in  Draco  and  would  be  the  same  star  as  Y 
Draconis. 

The  first  method  is  the  one  in  most  general  use.  It  seems  to 
be  the  one  most  analogous  to  the  ordinary  method  of  naming 
stars.  It  is  interesting  to  note  that  quite  recently  Nijland  has 
suggested  the  adoption  of  Andre's  nomenclature,  on  account 
of  the  possibility  of  its  unlimited  extension.1 

There  are  two  principal  sources  of  information  in  general  use 
at  the  present  time  regarding  variable  stars,  in  which  the  data 
are  arranged  in  catalogue  form :  the  Harvard  Catalogue  of  Vari- 
able Stars  found  in  Annals,  H.C.O.,  55,  and  the  Katalog  und 
Ephemeriden  verdnderlichen  Sterne,  published  yearly  by  Hart- 
wig  in  the  Vierteljahrsschrift  der  Astronomischen  Gesellschaft, 
and  obtainable  in  a  separate  pamphlet  on  request.  The  Har- 
vard catalogue  contains  the  following  information,  the  stars 
being  arranged  in  order  of  right  ascension :  the  Harvard  number, 
the  common  name  of  the  star,  the  number  in  the  Durchmuster- 
ung  zone,  the  right  ascension  and  decimation  for  1900,  the 
magnitude  at  maximum  and  minimum,  the  length  of  the 
period,  the  epoch  expressed  in  Julian  Days,  the  class  of  vari- 
able, the  type  of  spectrum,  the  year  of  discovery  with  the 
provisional  number,  and  the  name  of  the  discoverer.  The  table 
is  followed  by  copious  notes.  Another  table  in  this  important 
and  valuable  publication  contains  many  miscellaneous  facts, 
such  as  the  color,  the  interval  of  time  "maximum  minus  mini- 
mum," and  a  bibliography  of  maps  on  which  the  variable  is  to 
be  found.  Since  the  spectrum  of  a  variable  is  a  fact  of  extreme 
importance,  it  may  be  stated  here  that  a  more  complete  table 
containing  this  information  regarding  a  great  number  of  vari- 
ables is  to  be  found  in  Annals,  H.C.O.,  vol.  56,  no.  6. 

The  Ephemeriden  published  by  Hartwig,  as  the  name  implies, 
is  intended  for  practical  use  in  making  up  observing  lists  and 
for  the  purpose  of  comparing  observed  maxima  and  minima 

*  A.N.  4765. 


CATALOGUES  OF  VARIABLES  71 

with  the  computed  times.  In  describing  it  the  volume  for  1914 
will  be  used  as  a  type.  The  first  pages  are  devoted  to  an  intro- 
duction, which  contains  a  statement  of  the  changes  which  have 
been  made  in  the  elements  of  the  variables  already  known  and 
the  introduction  of  new  ones.  This  particular  number  contains 
a  discussion  by  Hartwig  regarding  the  introduction  of  the  name 
"Blinkstern"  as  a  substitute  for  "Cepheid"  variable  which 
was  mentioned  in  the  first  chapter  of  this  book. 

The  ephemeris  proper  is  divided  into  four  parts.  Part  I  is 
further  divided  into  two  groups,  the  first  containing  those  vari- 
ables whose  declination  is  north  of  —23°  and  the  second  those 
which  are  south  of  it.  In  respect  to  their  arrangement  the  two 
divisions  are  the  same.  The  reason  for  the  separation  is  that 
the  Durchmusterung  maps  on  which  the  variables  are  to  be 
found  are  prepared  for  different  epochs,  that  for  the  northern 
stars,  which  is  Argelander's,  being  for  the  epoch  1855,  and  that 
for  the  southern  being  for  1875,  the  date  of  the  catalogue  of  the 
Cordoba  Observatory.  The  tables  contain  the  current  number, 
the  right  ascension  and  declination  for  the  epoch  of  the  cata- 
logue, with  the  annual  precession,  and  if  the  star  is  of  long 
period,  the  elements  of  variation,  which  include  the  epoch  and 
the  period,  with  the  addition  of  any  terms  indicating  periodic 
or  secular  change  in  the  elements.  In  the  column  for  the  ele- 
ments are  inserted  the  words  "unbekannt"  (unknown)  and 
"unregelmassig"  (irregular),  when  necessary.  If  a  star  is  not 
a  long  period  variable,  it  is  stated  in  which  of  the  following 
divisions  of  the  pamphlet  it  is  to  be  found;  e.g.,  SY  Cass.  is 
placed  in  Abt.  II;  TV  Cass.  in  Abt.  Ill,  2;  SX  Cass.  is  Abt.  IV, 
1.  On  the  page  facing  this  are  further  data  regarding  the  same 
stars;  viz.,  the  magnitude  at  maximum  and  minimum,  and  the 
date  for  each,  predicted  from  the  elements  on  the  left  hand 
page.  Where  the  data  for  the  predictions  are  incomplete, 
blank  spaces  are  left  or  the  word  "Unbekannt"  is  inserted. 

Part  II  contains  the  following  data  regarding  short  period 
variables:  the  name  of  the  star;  the  phase  represented,  whether 
maximum  or  minimum;  the  epoch  in  Julian  Days,  the  number  of 


72  THE  STUDY  OF  VARIABLE  STARS 

decimal  places  in  which  expresses  the  accuracy  of  the  deter- 
mination; the  period  in  days;  the  quantity  M-m  when  known, 
and  the  type  of  variation;  i.e.,  whether  the  star  be  a  "Blink- 
stern,"  or  of  the  £  Geminorum  type,  or  belong  in  some  other 
group.  At  the  end  of  the  section  there  are  placed  in  one  group 
the  stars  from  this  table  which  have  periods  less  than  one  day 
and  hence  change  with  great  rapidity.  For  these  stars  the  right 
ascension,  declination,  and  longitude  for  1900  are  given,  also  a 
quantity  which  is  used  in  obtaining  the  reduction  to  the  sun,  a 
correction  which  is  usually  applied  to  the  observed  time  of 
maximum  or  minimum  when  the  period  is  very  short  and 
regular.  A  full  explanation  of  the  formula  and  its  application 
will  be  given  in  the  chapter  on  prediction. 

Part  III  contains  the  heliocentric  minima  of  the  Algol  stars 
computed  for  Greenwich  Mean  Time.  The  stars  are  arranged 
in  order  of  right  ascension.  For  each  is  given  the  elements  with 
the  name  of  the  authority.  When  the  period  is  very  short,  one 
minimum  for  each  month  is  given  and  multiples  of  the  period 
are  tabulated  by  the  application  of  which  the  other  minima 
may  be  found.  Following  this  material  is  a  table  which  con- 
tains the  data  for  the  reduction  to  the  sun,  and  also  the  dura- 
tion of  phase,  that  is,  the  time  from  maximum  through  mini- 
mum to  maximum,  or  the  period  of  time  during  which  the  star 
is  varying.  This  is  marked  D.  The  column  d  contains  the 
length  of  time  during  which  the  star  is  at  minimum.  Some- 
times this  is  very  short,  as  in  the  case  of  Algol,  and  sometimes 
it  lasts  for  an  hour  or  more,  as  in  the  case  of  U  Cephei. 

In  the  publications  previous  to  1914,  Table  IV  contained  the 
necessary  data  for  the  ant-Algol  stars,  but  in  that  for  1914 
these  stars  are  combined  with  the  other  Blink  stars,  and  Table 
IV  is  reserved  for  the  ft  Lyrae  stars.  The  arrangement  of  data 
is  practically  the  same  as  for  the  Algol  stars.  Following  this  is 
a  table  giving  a  key  which  will  enable  one  to  find  the  number 
of  the  variable  in  the  catalogue  from  its  letter. 

The  value  of  this  Ephemeris  cannot  be  overestimated,  and  it 
is  quite  indispensable  to  a  worker  in  variable  stars. 


CATALOGUES  OF  VARIABLES  73 

Several  important  catalogues  which  appeared  previously  to 
the  publication  of  the  Provisional  Catalogue  of  the  Harvard 
Observatory  were  prepared  by  S.  C.  Chandler,  the  first  one 
appearing  in  the  Astronomical  Journal  for  September,  1888.1 
According  to  Chandler's  own  statement  the  catalogue  was  not 
a  mere  compilation,  but  at  its  publication  involved  the  collec- 
tion of  all  the  published  observations  of  the  known  variables 
since  their  discovery,  including  his  own  unpublished  results, 
which  related  to  nearly  the  whole  list  of  variables  visible  in  the 
latitude  of  Boston.  A  discussion  more  or  less  complete  of  this 
material  furnished  the  values  of  the  elements  of  the  light  vari- 
ations in  his  catalogue.  In  it  he  first  introduced  his  method  of 
numbering  the  variables,  described  on  an  earlier  page  of  this 
chapter.  The  first  catalogue  contains  the  following  information : 
the  Chandler  number,  the  number  in  a  former  catalogue  pub- 
lished by  Schonfeld,  the  position  for  1855  with  the  precession, 
the  name  of  the  discoverer  with  the  date,  the  redness  of  the 
star,  the  magnitude  at  maximum  and  minimum,  the  Greenwich 
Mean  Time  of  the  epoch  either  maximum  or  minimum,  the 
length  of  the  period,  remarks,  and  the  position  for  1900.  The 
third  catalogue,  published  in  1893,  contains  important  material 
in  the  same  line,  with  very  few  changes,  except  that  the  Julian 
Day  of  the  epoch  is  added  to  the  calendar  date.  Periodic  in- 
equalities are  several  times  given  with  elements.  This  cata- 
logue was  considered  the  standard  until  the  publication,  as 
before  stated,  of  the  Harvard  catalogue. 

Since  the  data  connected  with  the  discovery  of  variables  are 
included  in  several  catalogues,  this  seems  a  suitable  place  for 
giving  some  account  of  the  methods  by  which  they  have  been 
discovered,  especially  of  some  of  the  systematic  searches  which 
are  being  made  for  them. 

The  methods  in  which  the  variability  of  a  star  is  discovered 
may  be  roughly  classed  as  visual,  photographic,  and  spectro- 
scopic.  The  discovery  may  be  made  visually  in  several  differ- 
ent ways,  by  direct  observation  either  with  the  naked  eye  or 
1  Ast.  Jour.,  8,  81. 


74  THE  STUDY  OF  VARIABLE  STARS 

with  the  telescope  in  the  process  of  making  star  catalogues, 
through  an  investigation  of  missing  DM.  stars,  and  as  the  result 
of  the  observation  of  comparison  stars  for  other  variables. 
These  four  methods  may  be  easily  illustrated.  The  earliest 
known  variable,  Mira  Ceti,  was  discovered  because  at  its  maxi- 
mum brightness  it  was  a  reddish  star,  rather  conspicuous,  and 
at  its  minimum  either  invisible  or  else  extremely  faint.  It  was 
first  noticed  by  Fabricius  in  1596,  and  again  in  1638  by 
Holwarda,  whose  observation  of  it  may  be  described  in  the 
following  terms  i1  on  the  16th  of  December,  1638,  he  was  occu- 
pied in  measuring  the  altitude  of  the  stars  above  the  horizon 
through  a  cloudy  sky,  for  the  purpose  of  observing  an  eclipse 
of  the  moon,  when  he  saw  three  times  something  very  brilliant 
and  new  sparkling  in  the  constellation  of  Cetus;  but  as  he  was 
preoccupied  with  the  eclipse,  he  did  not  disturb  himself  further 
in  regard  to  it.  However,  some  days  later  he  was  again  observ- 
ing the  sky  for  the  purpose  of  verifying  the  altitudes  previously 
determined,  when  his  eye  fell  by  accident  upon  Cetus.  He 
again  saw  shining  something  unfamiliar  to  him.  He  supposed 
this  apparition  to  be  a  temporary  meteor,  but  the  following 
day,  according  to  the  advice  of  the  Professor  of  Mathematics, 
he  undertook  the  examination  of  that  portion  of  the  sky.  The 
apparition  which  had  so  impressed  him  was  still  visible,  and  he 
began  to  study  it  attentively  and  to  determine  its  position. 
Both  to  the  naked  eye  and  to  the  telescope  it  resembled  the 
other  stars.  It  was  brighter  than  neighboring  stars  of  the  third 
magnitude.  Some  time  later  he  was  no  longer  able  to  find  this 
star,  and  he  believed  that  it  had  disappeared;  but  great  was  his 
astonishment  when  on  the  7th  of  November,  1639,  he  perceived 
it  hi  the  same  place  which  it  had  occupied  at  its  first  appear- 
ance, and  shining  with  a  brightness  sensibly  the  same.  At  this 
epoch  the  star  was  considered  by  astronomers  as  a  new  star, 
but  Holwarda  took  particular  pains  to  show  that  it  had  been 
known  for  a  long  time  and  that  Bayer  had  catalogued  it  in  1603 
as  being  of  the  fourth  magnitude,  under  the  name  of  o  Ceti. 
1  Ch.  Andr€.  Traitt  d' Astronomic  Stellaire,  I,  300. 


CATALOGUES  OF  VARIABLES  75 

Its  identification  led  to  consequences  of  the  greatest  impor- 
tance, proving  its  rapid  and  periodic  variability,  and  this  fact 
seemed  so  remarkable  that  Holwarda's  contemporaries  gave 
to  this  star  the  name  Mira  Ceti,  the  wonderful. 

The  star  Algol  was  also  one  of  the  earliest  variables  to  be 
discovered,  having  been  known  since  1669.  Its  remarkable  vari- 
ation won  for  it  its  name,  which  signifies  the  "Demon  Star." 

The  following  extract  from  a  letter  written  by  John  Good- 
ricke,  June  27,  1785,  and  printed  in  the  Philosophical  Trans- 
actions 1  for  that  year,  describes  his  discovery  of  the  variability 
of  /3Lyrae:  — 

The  account  that  has  been  lately  given  of  the  regular  variation  of 
Algol's  light  and  the  notice  astronomers  have  been  pleased  to  take  of 
it,  are  well  known.  It  is  natural  therefore  to  suppose,  that  the  rela- 
tion of  other  similar  phaenomena  may  also  meet  with  the  same  favor- 
able reception.  Of  this  kind  is  the  following,  which  I  beg  the  favor  of 
you  to  present  to  the  Royal  Society. 

On  the  10th  of  September,  1784,  whilst  my  attention  was  directed 
towards  that  part  of  the  heavens  where  /3  Lyrae  was  situated,  I  was 
surprised  to  find  this  star  much  less  bright  than  usual,  whereupon  I 
suspected  that  it  might  be  a  variable  star:  my  suspicions  were  after- 
wards confirmed  by  a  series  of  observations,  which  have  been  regu- 
larly continued  since  that  time,  and  which  will  presently  follow  in  their 
proper  place.  At  first  I  thought  the  light  of  this  star  subject  to  a 
periodical  variation  of  nearly  six  days  and  nine  hours,  though  the 
degree  of  its  diminution  did  not  then  appear  to  be  constant;  but  now 
upon  a  more  close  examination  of  the  observations  themselves,  I  am 
inclined  to  think  that  the  extent  of  its  variation  is  twelve  days 
and  nineteen  hours,  during  which  time  it  undergoes  the  following 
changes.  .  .  . 

A  person  who  is  habitually  interested  in  observing  the  lucid 
stars  acquires  a  great  familiarity  with  the  aspect  of  the  con- 
stellations and  can  quickly  discover  an  object  of  unusual  inter- 
est. The  two  brightest  novae  of  recent  years  were  discovered 
in  this  way  by  the  same  person,  Dr.  Thomas  Anderson,  who 
was  the  first  to  observe  Nova  Aurigae  and  Nova  Persei.  He 
stated  that  if  he  were  to  see  in  a  constellation  an  additional  star 
1  Phil.  Trans.,  75, 153. 


76  THE  STUDY  OF  VARIABLE  STARS 

as  bright  as  the  third  magnitude,  he  would  recognize  it  imme- 
diately as  a  new  star. 

Many  variables  have  been  discovered  in  the  course  of  making 
star  catalogues.  An  important  factor  hi  the  identification  of  the 
star  is  its  magnitude,  hence  in  preparing  the  catalogue  the 
observer  compares  his  observation  of  magnitude  with  those  of 
previous  observers,  and  thus  any  decided  difference  will  be 
detected.  Sometimes  his  individual  observations  will  differ 
among  themselves  to  such  an  extent  that  the  star  is  suspected 
of  variability.  Sometimes  in  using  the  ED.  catalogue  a  star 
will  seem  to  be  lacking.  On  investigation  it  may  turn  out  to 
be  a  variable,  or  an  error  in  the  original  observation  may  be 
discovered  by  referring  to  the  records  which  were  preserved  in 
the  library  of  the  Bonn  Observatory  for  this  very  purpose, 
according  to  Argelander's  expressed  wish.  The  following  illus- 
trations may  be  of  interest. 

Here  is  an  account  of  the  discovery  by  Espin  of  a  variable 
star,  which  is  announced  in  the  Astronomische  Nachrichten  3264. 
While  observing  a  star  for  his  catalogue  he  found  that  the  mag- 
nitudes, as  observed  on  four  nights,  were  8.6,  9.0,  9.6,  9.8. 
These  showed  that  the  star  was  a  variable,  which  fact  was  later 
confirmed,  and  it  is  now  known  as  W  Cassiopeiae.  In  Astro- 
nomische Nachrichten  3269  de  Ball  writes  that  at  the  Von 
Kuffner  Observatory  in  Vienna  he  was  not  able  to  see  a  star 
BD—6  5419  in  his  meridian  telescope  of  4.5  inches  aperture, 
but  found  another  faint  star  9.8  nag.  near  by.  Some  days  later 
he  observed  the  same  region  again  and  saw  both  stars,  the  miss- 
ing star  having  reached  9.0  mg.  Since  he  could  no  longer 
observe  the  region  with  the  meridian  instrument,  he  asked  Dr. 
Holetschek  to  follow  it  with  the  equatorial.  The  latter  did  so 
and  confirmed  the  variability  of  the  star,  which  is  now  known 
as  Z  Aquilae.  A  communication  from  Kustner  states  that  the 
star  had  been  observed  twice  by  Argelander  in  forming  the 
Durchmusterung,  with  the  magnitudes  9.0  and  9.3. 

The  circumpolar  variable  U  Cephei  was  observed  several 
times  by  different  astronomers  in  the  process  of  making  star 


CATALOGUES  OF  VARIABLES  77 

catalogues.  Each  time  it  was  seen  near  its  maximum  bright- 
ness, 6.9  mg.,  excepting  once  when  it  was  found  by  Schwerd1 
to  be  10.0  mg.,  but  this  discrepancy  was  not  noticed  until  the 
variability  of  the  star  had  been  discovered  by  the  Ceraskis 
from  the  examination  of  photographs.  The  star  TY  Aquilae, 
which  is  one  of  the  comparison  stars  for  W  Aquilae,  was  dis- 
covered to  have  a  small  variation  by  an  observer2  who  was 
following  W  Aquilae,  and  the  variation  was  later  confirmed 
by  E.  C.  Pickering. 

An  interesting  case,  and  one  not  so  satisfactory,  is  that  of 
BD  22°  3272,  to  which  attention  is  called  by  Becker  in  Astro- 
nomische  Nachrichten  3281  as  being  a  suspicious  object.  While 
observing  with  the  meridian  instrument  he  found  that  the 
above  star,  which  has  magnitude  8.9,  was  not  visible  in  the 
Berlin  transit  instrument  of  seven  inches  aperture,  while 
another  star  near  by  was  visible.  Neither  could  the  missing 
star  be  found  in  the  eighteen-inch  refractor  of  the  Strassburg 
Observatory.  On  the  other  hand,  a  letter  from  Professor  Kiist- 
ner  at  Bonn  stated  that  the  region  had  been  observed  three 
times  in  the  process  of  forming  the  BD.  catalogue,  and  the 
original  records  showed  that  undoubtedly  the  two  stars  were 
then  visible.  The  Harvard  Catalogue  of  Variable  Stars,  in  its 
remarks  concerning  this  star,  which  is  called  RW  Herculis, 
states  that  a  faint  object  which  had  been  observed  near  its 
position  by  Kobold  had  not  been  found  to  vary,  and  that 
eleven  photographs  taken  between  the  years  1890  and  1904 
showed  only  a  very  faint  star  with  no  certain  variation.  Visual 
estimates  of  the  magnitude  made  at  Harvard  showed  it  to  be 
12.0  mg.,  without  any  sure  evidence  of  change. 

An  interesting  and  almost  unique  instance  of  the  discovery 
of  a  variable  by  the  meridian  photometer  is  mentioned  by 
Professor  Pickering  in  the  Harvard  Circular,  No.  87.  While 
working  with  the  twelve-inch  meridian  photometer  he  ob- 
served a  bright  star  having  magnitude  9.5,  which  was  not 
in  the  BD.  An  examination  of  the  photographs  of  this  region 

1  S.  C.  Chandler,  Ast.  Jour.,  9,  49.       2  C.  E.  Furness,  Ast.  Jour.,  26,  74. 


78  THE  STUDY  OF  VARIABLE  STARS 

showed  that  the  star  was  a  variable  of  long  period.  This  is 
now  SV  Herculis. 

In  Astronomische  Nachrichten  3219  there  is  an  example  of 
the  discovery  of  a  variable  by  the  comparison  of  photographs. 
Announcement  was  made  from  the  Cape  of  Good  Hope  by 
Gill  that  a  comparison  of  two  photographic  plates  showed  a 
marked  difference  in  the  images  of  a  star,  which  was  later  con- 
firmed by  the  examination  of  a  number  of  other  plates.  This 
star  is  S  Velorum. 

A  very  interesting  and  rapid  method  of  discovering  variables 
is  by  the  use  of  the  stereo-comparator.  This  instrument  was 
used  by  Wolf 1  at  the  Heidelberg  Observatory  for  an  examina- 
tion of  the  region  of  the  nebula  of  Orion,  during  the  process  of 
which  he  discovered  ten  new  variables.  A  brief  description  of 
the  properties  of  this  instrument  will  show  how  easily  it  can 
be  used  for  the  discovery  of  variables.  It  is  constructed  so 
that  the  light  from  two  photographic  plates  can  be  thrown  into 
the  eyepiece  by  the  means  of  several  totally  reflecting  prisms. 
The  light  from  either  plate  can  be  shut  off  at  will,  so  that  the 
observer  can  use  first  one,  then  the  other,  or  both  together,  by 
rapidly  moving  the  shutter  by  which  this  is  accomplished.  He 
can  then  easily  determine  whether  the  star  images  on  both 
plates  are  exactly  the  same.  If  a  star  appears  on  one  plate  and 
not  on  the  other,  while  other  stars  are  alike  on  both,  the  sup- 
position is  that  it  may  be  a  variable.  The  same  region  was 
investigated  at  the  Harvard  Observatory 2  by  the  comparison 
of  photographic  plates,  and  several  of  Wolf's  variables  were 
confirmed,  while  many  new  ones  were  discovered.  This  led 
Professor  Pickering  to  arrange  for  a  systematic  study  of  the 
Orion  region  and  similar  nebular  regions  in  the  sky.  His  method 
is  as  follows:  the  glass  positive  of  one  of  the  plates  is  made, 
upon  which  are  superposed  the  negatives  of  the  same  region. 
On  the  positive  the  star  images  are  white,  while  on  the  negative 
they  are  dark;  thus  a  dark  image  is  superposed  upon  a  white 
one,  and  any  disparity  in  size  will  be  detected.  In  this  way  all 
1  A.N.  3749.  8  E.  C.  Pickering,  H.C.O.,  Circ.,  78. 


CATALOGUES  OF  VARIABLES  79 

the  variables  which  show  striking  changes  and  are  compara- 
tively bright  are  discovered.  In  Circular  79  he  announced  the 
discovery  of  76  new  variables,  19  in  Orion  and  Carina,  and  57 
in  the  small  Magellanic  cloud.  Later  circulars  have  contained 
the  announcement  of  the  discovery  of  many  other  variables 
found  in  the  same  way.  The  plates  which  are  mentioned  were 
taken  with  long  exposures.  It  was  found  later  *  that  other 
plates  taken  with  a  small  telescope  which  covered  a  region  of 
the  sky  30°  square  and  show  stars  of  the  llth  magnitude  and 
brighter,  would  furnish  a  valuable  means  of  discovering  the 
brighter  variables.  The  same  method  which  has  just  been 
described  was  followed,  a  thin  positive  being  made  of  each 
region,  upon  which  the  negatives  were  superposed.  The  plates 
covering  the  heavens  were  divided  into  groups  and  placed  in 
the  hands  of  the  observing  corps  for  investigation.  The  num- 
ber of  variables  which  have  been  discovered  by  this  method  is 
now  quite  large. 

It  was  stated  that  a  number  of  variable  stars  had  been  dis- 
covered from  an  examination  of  the  spectrum.  In  Astronomische 
Nachrichten  3269  is  found  an  announcement  by  Mrs.  Fleming 
of  the  discovery  of  bright  hydrogen  lines  in  the  spectrum  of  a 
star  of  the  third  type.  Photographs  of  this  star  were  then 
examined  which  showed  a  variation  in  brightness,  thus  proving 
the  star  to  be  a  variable.  In  Astronomische  Nachrichten  3225, 
Pickering  announces  that  four  new  variable  stars  have  been 
discovered  from  the  presence  of  bright  hydrogen  lines  in  their 
photographic  spectra.  They  are  all  long  period  variables. 

Perhaps  the  most  interesting  case  is  the  discovery  of  a  new 
star  by  means  of  its  spectrum.  This  was  accomplished  by  Mrs. 
Fleming  from  the  examination  of  negatives  made  at  the 
Arequipa  Observatory.  The  photograph,  which  was  taken 
with  an  exposure  of  an  hour,  showed  the  peculiar  spectrum, 
typical  of  new  stars,  in  which  certain  of  the  hydrogen  lines 
were  bright  and  were  accompanied  by  dark  lines  of  slightly 
shorter  wave-length.  Another  plate  of  the  same  region  was 
1  E.  C.  Pickering,  H.C.O.,  Giro.  122  and  127. 


80  THE  STUDY  OF  VARIABLE  STARS 

examined,  which  showed  a  change  in  the  spectrum,  further 
confirming  the  supposition  that  it  was  a  new  star.  The  exami- 
nation was  next  made  of  all  of  fhe  photographs  of  the  region 
containing  this  star  on  a  series  of  62  plates  extending  from  May, 
1889,  to  March,  1895.  No  trace  of  the  star  was  visible,  al- 
though on  some  of  them  stars  as  faint  as  the  14th  magnitude 
were  clearly  seen.  Beginning  with  the  plates  taken  on  April  8, 
1895,  the  star  appeared,  and  its  photometric  brightness  dimin- 
ished from  that  time  from  the  8th  to  the  llth  magnitude. 

It  is  from  the  study  of  photographic  plates  also  that  vari- 
able stars  in  cluster  have  been  discovered  on  a  large  scale. 
This  work  has  been  done  at  the  Harvard  Observatory  in  con- 
nection with  the  station  in  Arequipa,  Peru,  from  which  many 
photographs  were  taken  of  dense  globular  clusters  in  the  south- 
ern part  of  the  heavens.  The  process  is  related  at  length  in 
volume  38  of  the  Annals,  in  which  Bailey  describes  the  work 
on  the  cluster  Omega  Centauri.  In  the  introductory  pages  he 
gives  an  account  of  the  discoveries  which  led  to  a  thorough 
investigation  of  this  cluster  and  other  similar  ones.  The  first 
of  these  variables  was  found  by  Pickering  in  1889,  in  Messier  3, 
and  in  1890  two  other  variables  were  discovered  near  the  clus- 
ter Messier  5.  Later  in  the  same  year  the  variability  of  several 
other  stars  near  the  same  cluster  was  detected.  In  1893  a  vari- 
able star  was  discovered  on  the  photographs  of  Omega  Centauri 
by  Mrs.  Fleming,  and  a  few  days  later  another  by  Pickering. 
The  detection  of  these  variables  among  the  clusters  led  to  a 
systematic  examination  at  Arequipa  of  the  finest  globular  clus- 
ters for  the  discovery  of  new  variables.  The  first  objects 
examined  were  Messier  5  and  Messier  3,  both  of  which  gave 
remarkable  results.  A  table  is  presented  showing  the  number 
of  stars  examined  in  each  cluster,  and  the  number  of  variables 
found,  making  a  total  of  19,050  stars,  among  which  509  vari- 
ables were  found.  There  is  no  doubt  that  this  method  if  ex- 
tended to  all  of  the  clusters  in  the  sky  will  yield  important 
results.  The  instances  described  above  show  how  numerous 
the  methods  are  by  which  variable  stars  may  be  discovered. 


CHAPTER  V 

STELLAR  MAGNITUDE 

THE  history  of  stellar  magnitude  begins  with  the  catalogue 
of  Ptolemy,  which  is  the  oldest  that  has  been  handed  down  to 
us,  and  on  account  of  the  importance  and  age  of  this  work  we 
may  appropriately  give  an  extended  account  of  it.  Ptolemy's 
catalogue  is  contained  in  the  seventh  and  eighth  books  of  his 
celebrated  treatise  on  Astronomy  called  the  MeydXrjs  JLvv. 
Ta£ea>9,  to  give  it  the  Greek  name,  or  the  Almagest,  if  we  use 
the  more  familiar  name,  which  the  Arabian  astronomers  ob- 
tained by  prefixing  the  Arabic  article  al  to  the  Greek  word 
/i€7tc7T05.  The  date  of  the  catalogue  is  138  A.D.  Several 
manuscripts  have  been  preserved  for  us  in  Greek,  Latin,  and 
Arabic,  the  best  and  most  perfect  being  the  Greek,  in  the  Na- 
tional Library  in  Paris.  The  best  printed  edition  is  that  of 
M.  1'Abbe*  Halma,  published  in  Paris  in  1816,  having  the  Greek 
text  on  one  page  or  column  and  the  French  translation  on  the 
adjacent  one.  The  catalogue  contains  1028  stars,  which  are 
divided  into  49  constellations.  As  arranged  by  Ptolemy  it  con- 
sists of  four  columns;  the  first  containing  the  name  of  the  star 
in  the  constellation;  the  second  its  longitude;  the  third  the 
latitude,  and  the  fourth  the  magnitude.  As  is  probably  well 
known  to  the  reader,  the  ancient  and  mediaeval  astronomers 
always  designated  a  star  by  its  position  in  the  figure  represent- 
ing the  constellation.  Not  all  of  the  stars,  however,  fell  in  the 
regions  occupied  by  the  figures,  hence  they  were  placed  by 
themselves  at  the  end  of  the  group  near  which  they  were  situ- 
ated and  were  called  a/JLoptycoroi,  or  unformed.  There  are  102 
of  these. 

The  stars  and  figures  were  habitually  drawn  on  globes  by 
the  ancients.  On  the  title-page  of  the  second  volume  of  Raima's 
edition  of  Ptolemy  is  an  illustration  supposed  to  represent  the 


82  THE  STUDY  OF  VARIABLE  STARS 

globe  of  Hipparchus.  A  very  good  picture  of  the  globe  used  by 
Tycho  Brahe  is  given  in  the  memorial  volume  Tychonis  Brake 
Astronomiae  instauratae  Mechanica,  prepared  by  Hasselberg,  in 
1901,  on  the  three  hundredth  anniversary  of  Tycho's  death. 
The  longitudes  are  given  in  degrees,  counted,  however,  not 
from  0°  to  360°,  but  according  to  the  sign  of  the  Zodiac  in 
which  the  star  occurs,  each  sign  containing  30°;  e.g.,  the  star 
Castor  has  longitude  "Gemini  23j°"  and  since  Gemini  is  the 
third  sign,  this  is  equivalent  to  83|°.  Since  in  Greek,  as  in 
Latin,  numbers  were  represented  by  letters  of  the  alphabet, 
the  magnitudes  are  called  a,  /3,  7,  8,  e,  f.  It  would  appear  from 
this  that  Ptolemy  did  not  recognize  fractional  magnitudes, 
but  we  find  that  he  was  aware  of  the  gradations  in  brightness 
between  the  whole  magnitudes,  for  the  words  /-tet'fow  and 
eXda-a-cov  were  occasionally  attached  to  the  letters  representing 
the  magnitudes,  the  former  to  show  that  a  star  was  brighter 
than  the  average  star  of  its  magnitude  and  the  latter  that  it 
was  fainter,  thus  in  reality  giving  the  magnitudes  to  thirds. 
This  fact  is  of  great  importance  in  the  study  of  magnitude  if 
we  are  to  consider  these  earliest  estimations  of  scientific  value 
in  determining  changes  in  brightness,  and  it  is  necessary  to 
investigate  carefully  what  Ptolemy's  divisions  stand  for  accord- 
ing to  modern  standards.  This  has  been  done  very  thoroughly 
by  Dr.  C.  S.  Peirce  and  the  results  published  in  the  Annals, 
H.C.O.,  9.  He  examined  first  the  various  manuscripts  in  order 
to  compare  the  readings  for  pei&v  and  eXao-cr&w,  which  may  be 
abbreviated  m  and  e  respectively,  and  formed  a  table  con- 
taining the  magnitudes  for  Ptolemy's  groups  1,  le,  2m,  2e,  2m, 
etc.  Later  in  Annals,  14,  another  comparison  of  757  of  these 
stars  was  made  by  Pickering,  and  the  magnitudes  obtained 
according  to  the  Harvard  Photometry.  They  are  as  follows:1 

Ptolemy:    1       le     2m   2    ?  2e     3m    3       3e     4m   4       4e     5m    5       6 
Harvard:    0.5    1.2    1.2    2.1    2.6    2.7    3.3    3.8    3.8    4.4    4.6   4.7    5.0    5.4 

Ptolemy's  magnitudes  were  adopted  without  revision  by 
astronomers  who  followed  him  except  in  the  case  of  Al  Sufi  the 
1  Annals,  14,  343. 


ID  ANN  IS    BAYER! 

K  HAINAN  I    "I     C 


VR  A  N  O;«* 
METRIA 


OMNIVM      AS  TERI5MOKVM 

COM  IN  ENS    SCHEMATA. 
NOVA     METHOOO 

DELJNE  ATA 
AF,RLI<     I.AMJM.S     EXPRES5A 


Plate  III 

(VRANO  METRIA)  FRONTISPIECE   FROM   BAYER'S   URANOMETRIA,  1639 


STELLAR  MAGNITUDE  83 

Persian  astronomer,  living  in  the  tenth  century,  who  revised 
them  with  much  care.  Ptolemy's  magnitudes  were  used  by 
Ulugh  Beigh  in  a  catalogue  published  about  1437  and  by 
Tycho  Brahe,  whose  catalogue  of  777  stars  was  published  in 
1602. 

The  next  important  advance  in  the  cataloguing  or  charting 
of  stars  is  connected  with  the  name  of  Bayer.  It  has  been 
stated  that  the  stars  were  always  depicted  on  solid  globes  which 
represented  the  outside  of  the  celestial  sphere,  and  the  maps 
also  were  so  drawn.  Bayer  in  1603  conceived  the  idea  of  print- 
ing the  stars  on  charts  which  represented  the  inside  of  the 
celestial  sphere,  thus  reversing  the  directions  east  and  west. 
This  was  a  great  convenience,  but  he  also  performed  another 
task  which  was  of  even  more  service;  viz.,  he  assigned  Greek 
letters  to  the  stars  as  names,  instead  of  continuing  the  clumsy 
notation  of  denoting  them  by  describing  their  positions  in  the 
figures  of  the  constellations.  He  used  as  a  basis  Tycho's  cata- 
logue of  777  stars,  to  which  he  added  500  more,  locating  the 
latter  not  by  direct  observation,  but  by  estimating  their  posi- 
tions from  the  other  stars.  It  is  no  surprise  to  us  to  learn  that 
the  innovation  of  reversing  the  direction  in  his  charts  of  the 
heavens  aroused  much  unfavorable  comment,  though  its  con- 
venience must  have  been  obvious.  There  is  some  question  as  to 
the  method  Bayer  used  in  assigning  his  letters,  but  the  state- 
ment is  that  he  gave  them  to  the  stars  in  each  of  Ptolemy's 
classes  of  magnitudes  in  the  order  in  which  they  occur  in  the 
constellation,1  somewhat  as  follows:  first  to  all  of  the  stars  in 
the  first  or  a  magnitude  in  the  order  in  which  they  occur  in  the 
figure  representing  the  constellation,  beginning  usually  with 
the  head;  second  to  all  of  the  stars  in  the  second  or  P  magnitude, 
and  so  on.  But  he  did  not  arrange  the  stars  in  the  entire  con- 
stellation in  their  order  nor  assign  the  names  to  them  on  that 
basis.  It  is  difficult  to  verify  this  statement,  since  early  editions 
of  his  charts  are  not  easy  to  find.  The  question  was  discussed 
at  length  by  Argelander  in  his  Defide  Uranometriae  Bayeri,  but 
1  Benjamin  A.  Gould,  Uran.  Arg.,  51. 


84  THE  STUDY  OF  VARIABLE  STARS 

the  present  writer  was  unable  to  procure  a  copy  of  this  work 
for  comparison.  The  following  table  is  of  interest,  because  it 
gives  the  stars  in  the  constellation  of  Gemini  taken  from  Tycho 
Brahe's  catalogue,  with  the  letters  inserted  by  Baily  as  given 
by  him  in  a  valuable  paper  in  the  Mem.  R.A.S.,  13.  The  accom- 
panying plates  show  the  title-page  of  an  edition  of  Bayer's  atlas 
printed  at  Ulm  in  1619,  and  one  of  the  constellations,  Gemini. 
It  is  pleasant  to  note  that  the  drawing  for  this  lively  pair  is 
from  the  hand  of  no  less  an  artist  than  Albrecht  Diirer,  who 
drew  figures  of  the  constellations  in  1515  which  were  in  general 
use  by  astronomers  and  map  makers.  This  one  and  several 
others  may  be  seen  depicted  on  the  ceiling  of  the  Grand  Central 
Station  in  New  York  City.  The  reader  is  advised  to  follow  the 
map  of  the  twins  star  by  star  along  with  the  names  of  Tycho. 
It  will  be  seen  that  there  is  not  strict  correspondence,  which  is 
probably  due  to  the  fact  that  we  have  not  placed  together  the 
combinations  which  Bayer  used.  The  title-page  is  well  worth 
study  merely  as  a  product  of  the  early  printer's  art.  The  Eng- 
lish of  the  following  list  is  taken  from  an  old  treatise  on 
Astronomy  published  by  Vincent  Whig  in  London,  1651,  and 
entitled  Harmonicon  Celeste. 

Mag.        Bayer. 

In  the  upper  head,  Castor,  Apollo.  2  a 

In  the  lower  head,  Pollux,  Hercules.  2  ft 

In  the  left  hand  of  the  former  twin.  5  0 

In  the  left  shoulder.  4  T 

In  the  shoulder  blade  of  the  same.  4  i 

In  the  right  shoulder.  5  v 

In  the  left  shoulder  of  the  following  twin.  4  K. 

In  the  right  side  of  the  former  twin.  '  6  A 

The  little  star  in  the  left  elbow  of  the  higher  twin.        6  b' 

In  the  northern  and  upper  knee.  3  e 

In  the  left  knee  of  the  following  twin.  3  £ 

In  the  belly  of  the  southern  twin.  3  8 

In  the  hamme  of  the  lower  twin.  4  A 

The  first  in  the  foot  of  the  former  twin.  4  rj 

The  following  star  in  the  same  foot,  call'd  the  heel.      3  /* 

In  the  end  of  the  right  foot  of  the  former  twin.  4  v 

The  light  star  of  the  foot.  2  y 


STELLAR  MAGNITUDE  85 

In  the  lower  foot  of  the  following  twin.  4  £ 

In  the  heel  of  the  same  foot.  6  e 

Above  the  knee  of  the  lower  twin.  6  d 

In  the  thigh  of  the  higher  twin.  6  w 

Beneath  the  head,  lower  in  the  hand.  6  <£ 

The  little  star  between  both  heads.  5  <r 

About  the  ear  of  the  higher  twin.  5  p 

The  former  at  the  top  of  the  foot.  4 

It  may  be  noted  that  Bayer  has  represented  the  twins  as 
facing  the  observer,  which  is  indeed  the  correct  way  to  place 
them  when  they  are  on  the  inside  of  the  celestial  sphere,  since 
on  the  globe  showing  the  outside  of  the  sphere  they  are  seen 
from  the  back.  This  latter  fact  can  be  verified  from  the  picture 
of  Tycho's  globe  found  in  his  book  of  instruments  mentioned 
above. 

The  first  important  work  of  recent  times  on  magnitude  was 
done  by  William  Herschel,  that  fertile-minded  and  original 
genius.  In  this  as  in  many  other  of  his  achievements  he  sug- 
gested the  idea  without  carrying  it  out  to  its  completion,  leav- 
ing for  others  the  opportunity  to  make  it  of  practical  and  imme- 
diate use.  His  first  paper  was  published  in  the  Philosophical 
Transactions,  76,  1796,  from  which  the  following  extracts  are 
taken.  The  title  of  his  paper  is  "  On  the  Method  of  observing 
the  Changes  that  happen  to  the  fixed  Stars;  with  some  Remarks 
on  the  Stability  of  the  Light  of  our  Sun.  To  which  is  added,  a 
Catalogue  of  comparative  Brightness,  for  ascertaining  the 
Permanency  of  the  Lustre  of  Stars."  He  begins  by  stating 
that  there  is  great  confusion  in  giving  magnitudes  to  stars, 
since  reference  is  made  to  an  imaginary  standard  which  is  the 
average  magnitude  of  a  class,  and  the  stars  are  not  compared 
with  one  another.  In  examining  them  he  found  that  a  star  in 
one  class  might  be  brighter  than  one  in  a  higher  class,  or  a  star 
of  so-called  fourth  magnitude  might  be  fainter  than  one  of  the 
fifth.  So  many  discrepancies  occurred  that  either  there  had 
been  many  changes  in  magnitude  since  the  time  of  Flamsteed, 
or  else  the  assignment  of  magnitudes  was  full  of  error.  At  that 
time  intermediate  magnitudes  between  whole  numbers  were 


86  THE  STUDY  OF  VARIABLE  STARS 

designated  thus:  1.2,  between  first  and  second,  but  nearer  first; 
2.1,  between  first  and  second,  but  nearer  second,  making  prac- 
tically a  division  into  thirds.  After  calling  attention  to  many 
suspicious  cases,  he  suggests  the  following  improvement  in  the 
method  of  notation.  Instead  of  giving  an  exact  magnitude  to  a 
star,  he  places  a  few  stars  in  a  series  based  upon  their  order  of 
brightness.  For  example,  CDE  signifies  that  D  is  intermediate 
in  brightness  between  C  and  E,  which  are  neighboring  stars 
not  too  different  from  D  in  brightness.  By  extending  the  series 
to  six  or  seven  stars,  the  brightness  of  any  one  star  becomes 
even  more  definitely  fixed.  He  then  further  developed  a  plan 
for  representing  by  arbitrary  symbols  certain  degrees  of  differ- 
ence between  pairs  of  stars.  The  reader  who  is  familiar  with 
Argelander's  step  method  of  comparing  variables,  will  recognize 
this  method  of  Herschel  as  its  prototype. 

When  two  stars  are  perfectly  alike  in  brightness,  so  that  by  looking 
often  and  a  long  while  at  them,  I  either  cannot  tell  which  is  the  bright- 
est, or  occasionally  think  one  the  largest,  and  sometimes,  not  long 
after,  give  the  preference  to  the  other,  I  put  down  their  numbers  to- 
gether, only  separated  by  a  point.  .  .  .  However,  it  can  happen  but 
very  seldom  that  the  equality  in  the  lustre  of  two  neighboring  stars  is 
so  perfect  as  not  to  leave  an  inclination  to  prefer  one  to  the  other; 
therefore  I  place  that  first  which  may  probably  be  the  largest,  even 
though  I  do  not  particularly  judge  it  to  be  so.  But  this  preference  is 
never  to  be  understood  to  extend  so  far  as  to  make  it  improper  to 
change  the  order  of  the  two  stars. 

Continuing  in  this  manner,  he  describes  his  other  steps,  and 
finally  introduces  a  table  of  symbols:  — 

*  The  least  perceptible  difference  less  bright. 
.    Equality. 

,    The  least  perceptible  difference  more  bright. 

-  A  very  small  difference  more  bright. 

-  ,     A  small  difference  more  bright. 

-  -    A  considerable  difference  more  bright. 

Any  great  difference  more  bright  in  general. 

When  two  stars  differ  so  much  in  brightness  that  one  or  two  other 
stars  might  be  put  between  them,  and  still  leave  sufficient  room  for 
distinction,  they  become  partly  unfit  for  standards  by  which  the  lustre 
of  other  stars  can  be  ascertained. 


STELLAR  MAGNITUDE  87 

There  exist  symbols  indicating  other  degrees  of  difference  in 
brightness,  but  enough  has  been  quoted  to  show  the  extent  to 
which  Herschel  had  developed  his  method.  He  has  also  a  series 
of  marks  which  indicate  a  wavering  of  star-light  and  describes 
quite  vividly  the  occasions  on  which  he  used  them. 

Sometimes,  when  I  was  not  willing  to  put  down  these  compound 
marks,  I  have  cast  my  eyes  upon  the  ground,  and  after  a  few  moments 
lifted  them  quickly  up  to  the  stars  AB,  and  instantly  decided  which  of 
the  expressions  ought  to  be  used;  this  being  repeated  perhaps  a  dozen 
or  more  times,  I  took  that  expression  for  the  most  proper  one  which 
would  occur  oftener  than  any  other  in  these  transitory  glances. 

He  also  calls  attention  to  various  subjective  errors  which  must 
be  avoided  in  making  these  comparisons. 

This  introduction  is  followed  by  the  First  Catalogue  of  the 
Comparative  Brightness  of  the  Stars,  which  contains  series  of 
stars  in  nine  constellations.  No  attempt  is  made  to  arrange 
them  all  in  order,  or  to  form  a  catalogue  of  magnitudes,  and 
only  the  separate  series  are  published,  which  furnish  material 
for  the  purpose  of  forming  a  catalogue.  Four  such  series  were 
published  by  Herschel.  Two  others  existed  in  manuscript  form 
and  were  given  into  the  hands  of  Professor  Pickering  for  publi- 
cation. An  investigation  of  them  is  given  in  Annals,  H.C.O., 
14,  where  the  values  of  the  three  symbols  most  frequently 
used  by  Herschel  are  found.  These  were  the  period,  the  comma, 
and  the  dash,  which  have  respectively  the  values  .06  mg.,  .23 
mg.,  .38  mg.  The  results  have  been  arranged  in  catalogue  form, 
and  are  published  in  Annals,  H.C.O.,  23,  188,  et  seq.,  Table 
LIV.  A  discussion  of  his  work  follows,  in  which  Pickering  states 
that  "Herschel  furnished  observations  of  nearly  three  thou- 
sand stars,  from  which  their  magnitudes  a  hundred  years  ago 
can  now  be  determined  with  an  accuracy  approaching  that  of 
the  best  modern  catalogues.  The  average  difference  from  the 
photometric  catalogues  is  only  ±0.16  mg.,  which  includes  the 
actual  variations  of  the  stars  during  a  century,  as  well  as  the 
errors  of  both  catalogues." 

We  now  come  to  the  great  work  of  Argelander  on  stellar 


88  THE  STUDY  OF  VARIABLE  STARS 

magnitude,  and  the  author  has  difficulty  in  deciding  how  much 
of  it  to  include  in  the  present  chapter.  However,  since  it  is  in 
the  BD.  catalogue  that  magnitudes  expressed  in  tenths  are 
first  introduced,  it  may  be  of  considerable  value  to  give  an  his- 
torical account  of  the^development  of  his  method.  The  mate- 
rial for  this  account  may  be  found  in  a  letter  from  Schonfeld 
to  C.  S.  Peirce l  written  about  twenty  years  after  the  comple- 
tion of  the  work. 

It  should  be  stated  as  a  preliminary  that  earlier  astronomers, 
beginning  with  Ptolemy,  recognized  the  fact  that  there  are 
magnitudes  intermediate  between  those  represented  by  whole 
numbers;  that  is,  that  the  eye  is  sensitive  to  smaller  differences 
in  brightness  than  are  indicated  by  the  ordinary  divisions. 
Ptolemy,  as  stated  before,  made  use  of  the  two  words  greater 
and  less  to  show  these  differences,  thus  practically  introducing 
thirds  of  magnitudes.  Later  astronomers,  among  them  Flam- 
steed  and  Herschel,  used  the  combinations  1.2  and  2.1  as  inter- 
mediate between  1  and  2,  meaning  that  1.2  was  brighter  than 
the  average  star  of  the  second  magnitude,  and  less  bright  than 
a  star  of  the  first,  but  nearer  in  brightness  to  the  star  of  the 
first  magnitude.  Similarly,  2.1  meant  that  the  star  was  inter- 
mediate in  brightness,  but  nearer  to  the  standard  second  mag- 
nitude star.  Thus  again  thirds  of  magnitudes  were  recognized. 
The  reader  is  particularly  cautioned  not  to  regard  the  periods 
in  the  expressions  1.2  and  2.1  as  decimal  points. 

Some  of  the  star  catalogues,  for  example,  that  of  Lalande, 
for  the  epoch  1800,  give  the  brightness  in  whole  and  half  mag- 
nitudes, as  6,  6j,  7,  7j,  etc.,  and  this  method  was  quite  custom- 
ary among  astronomers  when  Argelander  began  his  photometric 
work.  His  first  research  was  the  Uranometria  Nova,  published 
in  1843.  According  to  the  title-page  it  is  a  representation  of 
the  stars  visible  to  the  naked  eye  in  middle  Europe,  according 
to  their  true  magnitudes  taken  directly  from  the  sky.  It  con- 
sists of  a  series  of  seventeen  charts  and  a  small  book  containing 
a  description  of  the  charts  and  a  catalogue  of  the  stars.  It 
*  Annals,  H.C.O.,  Q,  27-28. 


STELLAR  MAGNITUDE  89 

served  as  a  model  for  the  work  of  Heis.  There  are  3256  objects 
delineated,  including  stars,  nebulae,  and  clusters.  There  are 
nineteen  degrees  of  brightness,  from  the  first  to  the  sixth  mag- 
nitude, including  three  for  each  class.  While  Argelander  does 
not  describe  in  detail  his  method  of  determining  these  magni- 
tudes, he  states  in  his  usual  vivid  style  that  he  began  his  task 
in  1838,  and  worked  on  it  diligently  until  the  time  of  publica- 
tion, making  repeated  comparisons  of  the  stars  among  them- 
selves. He  made  use  of  the  ordinary  six  classes,  calling  the  faint- 
est stars  which  he  could  see  distinctly  of  the  sixth  magnitude, 
saying  modestly,  "My  eye  is  of  ordinary  sharpness;  a  weaker 
one  will  not  see  the  smaller  stars  of  my  sixth  magnitude,  and 
a  stronger  one  will  perceive  many  which  remain  invisible  to 
me."  He  had  in  mind  not  only  the  professional  astronomer, 
but  also  the  amateur,  whom  he  calls  "der  Liebhaber  der 
Astronomic, "  and  says  that  he  aims  to  give  a  true  representa- 
tion of  the  relative  brightnesses  of  the  stars  of  his  own  time,  in 
order  that  his  successors  might  be  able  to  decide  whether  indi- 
vidual stars  had  changed  their  light.  Space  does  not  permit 
of  further  quotations,  but  there  is  a  charm  and  simplicity 
about  Argelander's  introductory  statements  which  make  them 
delightful  reading.  The  drawings  for  his  constellation  figures 
were  taken  from  Bayer's  Uranometria. 

We  pass  now  to  the  account  of  the  work  on  the  Durchmus- 
terung  magnitudes,  contained  in  Schonfeld's  letter  to  Peirce, 
mentioned  on  an  earlier  page.  The  zone  observations  for  the 
great  catalogue  were  made  with  a  small  telescope  of  three 
inches  aperture,  as  described  in  an  earlier  chapter,  while  the 
later  zones  for  the  purposes  of  revision  were  made  with  larger 
instruments.  The  catalogue  was  published  in  three  sections, 
and  the  method  of  estimating  the  magnitudes  in  them  changed, 
as  the  observers  gained  in  experience,  and  also  as  they  reached 
regions  of  the  sky  where  the  stars  were  fewer  in  number  and 
did  not  require  such  rapid  observing.  It  was  originally  the 
plan  to  estimate  the  brightness  in  half  magnitudes,  and  they 
adopted  the  scale  1  mg.,  1.5  nag.,  2  mg.,  2.5  mg.,etc.,  but  still 


90  THE  STUDY  OF  VARIABLE  STARS 

used  the  symbols  1.2  and  2.3  to  denote  the  magnitudes  between 
1  and  2,  2  and  3,  etc.,  which  in  this  case  were  the  exact  half 
magnitudes  1.5  and  2.5.  The  work  was  begun  in  1852,  and 
toward  the  end  of  the  year  1854  Schonfeld  and  Krueger,  Arge- 
lander's  assistants,  who  made  by  far  the  greater  part  of  the 
observations,  began  to  take  account  of  a  perceptible  difference 
from  a  half  magnitude;  for  example,  if  a  star  belonged  to  the 
faintest  among  those  classified  as  seventh  magnitude,  that  is, 
was  less  than  the  division  7.8  or  7.5  mg.,  it  was  distinguished 
by  the  addition  of  the  letter  s,  schwach  (faint),  or  7.8s,  which 
made  it  a  little  fainter  than  7.5  mg.,  or  the  equivalent  of  7.7  mg. 
Similarly  a  star  which  belonged  to  the  brighter  half  of  the  class 
was  designated  by  gt,  or  gut,  and  7.8gt  meant  a  star  somewhat 
brighter  than  7.5  mg.  or  7.3  mg.  Without  mentioning  further 
details,  it  is  sufficient  to  say  that  the  magnitudes  at  observa- 
tion were  divided  into  six  parts,  and  the  following  table  shows 
the  correspondence  between  their  designations  and  the  ordi- 
nary scale  of  tenths  of  magnitudes. 

7m  6.9,  7.0,  7.1  mg. 
7s  7.2 

7.8gt  7.3 

7.8  7.4,  7.5,  7.6 
7.8s  7.7 

8gt  7.8 

8  7.9,  8.0,  8.1 

In  explaining  why  three  tenths  were  included  in  one  group, 
Schonfeld  states  that  it  was  not  so  easy  to  make  the  distinction 
of  .1  as  of  .2  in  the  midst  of  observing;  i.e.,  to  distinguish  7.3 
from  7.5  was  easier  than  from  7.4,  and  besides,  the  stars  often 
came  so  rapidly  that  the  observers  had  no  time  to  write  the 
necessary  notes.  Hence  it  resulted  that  in  the  regions  where 
the  stars  were  fewer,  more  fine  differences  were  noted  than 
with  stars  in  the  Milky  Way. 

In  the  year  1857,  when  the  observers  had  reached  the  more 
northern  declinations,  and  the  stars  were  on  the  average  less 
numerous,  they  became  accustomed  to  distinguish  tenths  of 


STELLAR  MAGNITUDE  91 

magnitudes  directly.  But  it  was  noticeable  that  the  fractions 
.1,  .6,  and  .9  occurred  less  frequently  than  the  others,  especially 
.1  and  .6.  The  magnitudes  which  were  published  in  the  BD. 
are  the  means  of  the  separate  determinations,  some  of  them 
depending  on  two  observations  and  some  on  three.  When  they 
rested  on  two  observations,  the  tenth  was  chosen  so  that  the 
star  in  general  was  placed  fainter  than  the  arithmetical  mean, 
for  example,  if  a  star  at  one  observation  was  called  8.9,  which 
is  equivalent  to  8.5  mg.,  and  at  another  9,  the  mean,  which 
would  be  8.75,  was  called  8.8  and  not  8.7.  As  a  result  of  this 
variation  in  the  method  of  determining  the  magnitudes  of  the 
BD.,  it  appears  that  they  are  not  homogeneous  throughout  the 
entire  catalogue.  However,  in  spite  of  this,  they  are  very  much 
more  precise  than  those  found  in  any  of  the  preceding  cata- 
logues. 

Several  investigations  have  been  made  by  different  astrono- 
mers for  the  purpose  of  reducing  the  BD.  magnitudes  to  a  uni- 
form photometric  scale,  but  there  will  be  no  opportunity  for 
presenting  them  in  this  volume. 

While  working  on  the  catalogue,  Argelander  was  on  the  look- 
out for  possible  cases  of  variability.  In  the  earlier  zones  there 
were  frequently  rather  large  differences  between  the  estimates 
of  magnitude  in  the  different  zones,  but  later  this  seldom  hap- 
pened. If  any  such  did  occur,  the  star  was  at  once  investi- 
gated, and  if  the  divergences  persisted,  the  variability  of  the 
star  was  considered  as  fairly  well  established. 

The  remaining  extended  investigation  of  stellar  magnitude 
made  without  a  photometer  was  that  undertaken  by  Gould 
at  the  Cordoba  Observatory,  and  called  by  him  the  Urano- 
metria  Argentina.  The  introduction  contains  a  very  spirited 
account  of  his  work  and  the  difficulties  under  which  he  labored 
in  carrying  it  out.  The  last  paragraph  of  his  preface  shows  us 
how  deeply  influenced  he  was  by  the  spirit  of  Argelander :  — 

During  all  the  stages  of  this  undertaking,  and  the  not  small  dis- 
couragements which  have  attended  it,  I  found  incentive  and  support 


92  THE  STUDY  OF  VARIABLE  STARS 

in  looking  forward  with  hopefulness  to  the  approbation  of  the  great 
master  in  this  department  of  astronomy.  The  coveted  privilege  has 
not  been  granted  me,  to  lay  at  his  feet  the  finished  work.  But,  in 
justice  and  in  gratitude,  I  desire  to  record  my  obligations  to  him  for 
counsel  and  encouragement,  direct  and  indirect.  To  Argelander, 
living,  I  desired  to  inscribe  this  work,  which  but  for  his  Uranometria 
Nova  might  never  have  existed.  Now  I  may  only  dedicate  it  to  his 
honored  memory. 

Gould  had  been  summoned  by  the  Argentine  Republic  to 
establish  a  national  observatory  at  Cordoba,  several  hundred 
miles  inland  from  the  city  of  Buenos  Aires.  He  arrived  there 
during  the  year  1870  with  four  young  men  who  were  to  act  as 
his  assistants,  and  with  no  luggage  except  his  personal  belong- 
ings, the  war  then  raging  in  Europe  having  interfered  with  the 
shipment  of  boxes  containing  astronomical  books.  No  instru- 
ments were  at  hand  except  opera-glasses,  and  the  transit  instru- 
ment which  was  to  be  the  basis  of  the  principal  work  of  the 
observatory  was  not  put  in  place  for  nearly  two  years.  It 
appeared  to  him  that  the  energies  of  his  party  could  not  be 
better  employed  than  in  determining  the  relative  magnitudes 
of  the  southern  stars,  for  the  formation  of  an  Uranometry 
similar  to  that  of  Argelander.  By  means  of  a  star  catalogue 
which  he  had  with  him,  he  plotted  the  positions  of  the  brighter 
stars  upon  skeleton  maps,  and  inserted  the  fainter  stars  from 
direct  estimation  in  the  sky.  He  desired  to  base  his  scale  of 
magnitudes  as  far  as  possible  on  that  of  Argelander,  hence  he 
selected  as  a  standard  zone  the  belt  which  has  for  its  central 
line  a  declination  having  the  same  altitude  at  Cordoba  and 
Bonn.  This  was  +  9°  39'  15".  He  wished  to  express  his  own 
magnitudes  in  tenths,  but  as  the  Uranometria  Nova  gave  them 
only  in  thirds,  he  was  unable  to  make  more  than  a  general 
correspondence  between  the  two  scales.  The  stars  in  this  belt 
which  were  selected  as  standards  were  observed  by  all  of  his 
assistants;  only  those  were  utilized  upon  which  all  four  agreed, 
and  they  were  observed  most  elaborately  throughout  the  whole 
circumference  of  the  heavens.  During  this  process  he  discov- 


STELLAR  MAGNITUDE  93 

ered  that  the  scale  of  the  Uranometria  Nova  was  not  homoge- 
neous throughout  the  region  observed,  nor  did  Argelander  make 
very  precise  determinations  of  the  intermediate  grades  of  mag- 
nitude. He  also  found  that  in  the  clear  air  of  Cordoba  stars 
could  easily  be  seen  which  were  fainter  than  those  selected  as 
sixth  magnitude  at  Bonn.  He  finally  fixed  upon  7.0  mg.  as  the 
faintest  generally  visible.  There  were  722  standard  stars,  and 
the  entire  catalogue  contains  7730  south  of  declination  +10°. 
Besides  the  establishment  of  standard  magnitudes  Gould  found 
that  it  was  necessary  to  rectify  the  boundaries  of  the  southern 
constellations.  During  the  course  of  his  researches  he  found 
many  stars  which  showed  signs  of  variability,  and  made  them 
special  objects  of  observation.  Lack  of  time,  however,  pre- 
vented him  from  following  them  as  closely  as  he  would  have 
wished. 

Having  given  in  the  preceding  pages  a  general  survey  of  the 
earlier  work  on  magnitudes,  previous  to  the  introduction  of 
photometric  apparatus  and  the  more  exact  methods  of  the  last 
thirty  years,  there  remains  to  the  author  the  task  of  showing 
the  connection  between  the  relative  brightnesses  of  the  stars 
and  their  magnitudes.  While  it  would  be  interesting  histori- 
cally to  give  all  the  steps  which  led  to  the  formation  of  the 
present  formula,  it  is  possible  to  give  only  a  brief  resume",  and 
then  to  devote  some  time  to  examples  of  its  application.  The 
problem  may  be  stated  in  the  form  of  a  question,  the  necessity 
of  an  answer  to  which  will  be  obvious.  The  magnitudes  assigned 
to  the  stars  proceed  in  order  as  the  stars  diminish  in  bright- 
ness. Is  there  a  definite  relation  between  the  brightness  of  a 
star  of  one  magnitude  and  that  of  a  star  of  the  next  succeeding 
magnitude?  The  question  may  be  put  more  specifically.  In 
what  ratio  does  the  brightness  of  the  stars  change  as  we  pass 
from  the  first  to  the  second  magnitude  or  from  the  second  to 
the  third?  Is  there  any  definite  relation  or  is  it  just  a  matter  of 
chance?  Were  the  stars  divided  into  magnitudes  according  to 
some  general  scientific  scheme  or  was  it  merely  at  the  conven- 
ience of  the  early  observers?  The  answer  to  this  question,  in 


94  THE  STUDY  OF  VARIABLE  STARS 

whatever  form  it  may  be  put,  is  obviously  of  great  importance 
to  the  astronomer. 

Hipparchus,  127  B.C.,  was  the  first  to  form  a  star  catalogue 
and  assign  magnitudes  to  the  stars.  His  results  have  been 
preserved  to  us  by  Ptolemy,  but  we  are  not  aware  on  what 
principle  he  made  the  division.  Obviously  he  grouped  all  the 
brightest  stars  in  the  first  magnitude;  the  faintest  stars  he 
placed  in  the  sixth,  and  those  intermediate  in  brightness  he 
distributed  in  the  other  classes.  Whether  it  was  a  matter  of 
accident  that  he  chose  six  classes,  or  whether  he  had  a  definite 
idea  of  a  light  ratio  in  making  the  division  we  do  not  know. 
Ptolemy  is  silent  on  the  subject,  and  it  seems  probable  that  if 
there  had  been  a  governing  principle  he  would  have  stated  it. 
In  any  case,  for  centuries  the  magnitude  of  a  star  remained 
only  an  incidental  piece  of  information  to  assist  in  identifying 
it,  and  not  until  the  time  of  Herschel,  when  variation  in  stellar 
light  was  being  closely  studied  and  he  was  making  comparison 
of  the  brightness  of  the  stars,  did  the  matter  become  one  of 
importance.  With  the  growth  of  photometric  work  it  became 
necessary  to  investigate  this  relation,  with  the  result  that  we 
have  a  very  definite  formula,  known  as  Pogson's  rule,  which  is 
as  follows :  — 

Let  A  be  the  brightness  of  one  star, 

let  B  be  the  brightness  of  a  second  and  fainter  star, 

let  A  m  be  their  difference  in  magnitude; 

(1)  then  |  =  (2.512)Am- 

n 

If  we  place  A  m  =  1  we  find  that  a  star  of  one  magnitude  is 

2.512,  or  approximately  %\  times  as  bright  as  a  star  of  the  next 

lower  magnitude,  and  that  this  holds  everywhere  on  the  entire 

scale;  that  is,  there  is  a  constant  ratio  existing  between  the 

brightnesses  of  stars  of  successive  magnitudes.  A  few  facts  in 

the  history  of  the  derivation  of  this  number  may  now  be  given. 

Sir  John  Herschel1  was  one  of  the  first  to  formulate  some 

such  numerical  relation,  deducing  it  from  observations  made 

1  Mem.,  R.A.S.,  3, 182. 


STELLAR  MAGNITUDE  95 

at  the  Cape  of  Good  Hope.  His  results  were  somewhat  tenta- 
tive, and  he  concluded  that  the  quantities  of  light  emitted 
formed  a  series  of  inverse  squares,  such  as  1,  J,  £,  iV>  TS»  etc.; 
the  light  emitted  being  the  inverse  square  of  the  magnitudes, 
1,  2,  3,  4,  5,  etc.  At  another  time  he  said  that  he  thought  it 
better  in  the  case  of  stars  below  the  sixth  magnitude  to  halve 
the  light  of  each  magnitude  to  get  that  of  the  next  lower,  thus 
introducing  a  geometrical  ratio,  -J,  indicating  the  relative 
brightness  between  two  magnitudes,  as  y,  i,  J,  i\,  etc.  There 
seemed  no  desire  on  the  part  of  astronomers  to  disturb  the 
general  assignment  of  magnitudes  due  to  Ptolemy,  since  that 
would  involve  too  great  confusion. 

I  believe  this  principle,  which  assigns  a  decrease  of  light  in  geo- 
metrical progression,  according  to  the  powers  of  £,  while  the  order 
of  magnitude  increases  in  arithmetical,  is  preferable  to  that  which 
would  estimate  the  light  by  the  reciprocal  square  of  the  magnitude. 
.  .  .  From  some  experiments  I  have  made  with  apertures  of  various 
sizes  I  am  led  to  believe  that  the  actual  ratio  of  the  light  of  a  star 
of  the  first  magnitude  to  one  of  the  sixth  is  at  least  100:1 ;  for  I  found 
that  Sirius,  when  viewed  with  an  aperture  of  three  inches,  was  equal 
to  a  large  star  of  the  fourth  magnitude,  with  two  inches,  to  a  star  of 
the  4.5  magnitude,  and  with  one  inch  its  impression  on  the  eye,  in 
spite  of  the  large  planetary  disk  it  exhibited  under  those  circumstances, 
was  fully  equal  to  that  of  a  bright  star  of  the  sixth  magnitude  seen 
with  the  full  aperture  of  eighteen  inches,  which  would  give  324:1; 
and  admitting  Sirius  to  have  three  times  the  light  of  an  average  star 
of  the  first  magnitude,  we  get  the  ratio  above  stated. 

Other  workers  in  photometry  from  the  beginning  adopted 
the  geometrical  ratio  as  the  true  relation,  the  question  left  to 
them  being  to  find  its  value.  It  is  generally  known  as  p,  so  that 
the  formula  given  above  as  Pogson's  rule  should  read  in  general 

(2)  *~^"' 

Several  astronomers  and  physicists  began  working  on  the 
problem.  The  method  of  investigation  was  to  select  certain 
stars  the  magnitudes  of  which  had  previously  been  determined 
with  considerable  accuracy;  to  find  by  various  photometric 
means  their  actual  light  ratio,  and  then  by  using  the  above 


96  THE  STUDY  OF  VARIABLE  STARS 

equation  to  find  the  value  of  p.  Steinheil,  in  1835,  working  at 
Munich,  found  from  the  observation  of  thirty  stars  the  value 
of  p  to  be  2.831.  In  1851  Johnson,  at  Radcliffe,  found  from  the 
observation  of  sixty  stars  ranging  from  4.1  mg.  to  9.7  mg.  the 
ratio  of  diminution  to  be  0.424,  its  reciprocal  being  2.358.  He 
also  quotes  values  obtained  by  several  different  observers, 
using  instruments  varying  from  a  3.5-inch  to  a  15-inch  re- 
fractor and  an  18-inch  reflector.  "Yet  their  determinations, 
notwithstanding  many  and  great  individual  anomalies,  pre- 
sent a  general  appearance  of  consistence  and  agreement  which 
can  hardly  be  accidental."1 

It  may  be  interesting  to  note  them. 

Herschel  .407  Argelander  .431 

Struve  .383  Groombridge          .388 

OttoStruve          .406  Radcliffe  .424 

The  mean  of  these  is  .412,  or  p  =  2.427. 

Before  ref  erring  to  the  suggestion  of  Pogson,  which  led  to  the 
final  adoption  of  p  =  2.512,  it  will  be  convenient  to  transform 
equation  (2)  so  as  to  find  the  value  of  A  m,  as  follows: 


(3)  log  -  ==  A  m  log  p, 


It  will  be  seen  that  log  p  enters  as  a  divisor  in  the  equation 
when  Am  is  the  value  sought.  Let  us  now  introduce  a  few 
other  determinations  of  the  value  for  p  and  also  their  loga- 

rithms. 

P  logp 

Steinheil  2.831  0.4519       26  stars 

Johnson  2.427  0.3851        (mean) 

Stampfer  2.519  0.4012          132 

Pogson  2.4  0.3802 

Mean  0.4036 
1  Radcli/e  Obs.,  1851,  App.  25. 


STELLAR  MAGNITUDE  97 

Since  these  values  of  log  p  differ  considerably  among  them- 
selves, and  the  mean  value  is  .4036,  Pogson  decided  arbitrarily 
to  adopt  the  value  of  0.4,  on  account  of  its  convenience  as  a 
divisor,  thus  making  p  equal  to  2.512.  It  is  a  curious  coinci- 
dence that  this  determination  seems  to  have  been  entirely  unin- 
fluenced by  the  suggestion  of  Herschel  previously  quoted,  viz., 
that  a  star  of  the  first  magnitude  was  one  hundred  times  as 
bright  as  a  star  of  the  sixth,  for  proceeding  according  to  the 
hypothesis  that  the  brightness  changed  by  a  constant  ratio  we 
should  have  in  this  case 


A  m  =  5  mg.,  —  =  100, 


(4 


log  100    2 
log  P  =  -~~  «=-  =  .4, 


which  is  precisely  the  same  result.  On  account  of  its  general 
convenience  this  value  of  p  as  adopted  by  Pogson  has  found 
universal  acceptance.  A  few  later  investigators  have  made 
other  determinations  of  the  value  of  p,  and  efforts  have  been 
made  to  discover  if  it  really  is  constant,  and  expresses  a  law 
inherent  in  our  psycho-physical  natures,  but  a  discussion  of 
these  efforts  lies  somewhat  outside  the  scope  of  this  volume.  It 
should  be  remembered,  however,  that  in  the  end,  all  of  these 
determinations  lead  back  to  Ptolemy's  classification,  which 
was  handed  down  at  first  quite  unchanged,  and  later,  when 
improvements  were  found  necessary,  it  was  altered  only  by 
internal  adjustments,  its  outer  limits  remaining  fixed.  The 
brightest  stars  only  are  an  exception  to  this,  because,  in  order 
to  conform  to  the  general  light  ratio,  they  have  been  pushed 
out  of  the  first  magnitude,  as  it  were,  into  zero  or  negative 
magnitudes,  so  that  we  have  l 

1  Rev.  HP.t  Annds,  H.C.O.,  50,  237,  Table  vn. 


98 


THE  STUDY  OF  VARIABLE  STARS 


Sirius 

-1.58 

0  Orionis 

0.34 

Canopus 

-0.86 

Procyon 

0.48 

Vega 

0.14 

a  Eridani 

0.60 

a  Aurigae 

0.21 

/3  Centauri 

0.86 

Arcturus 

0.24 

Altair 

0.89 

a'  Centauri 

0.33 

Betelgeuse 

0.92 

The  statement  of  Johnson  that  this  agreement  of  results  for 
the  values  of  p  cannot  be  accidental  is  true,  for  all  of  the 
observers  used  the  same  scale  of  magnitudes  for  their  deter- 
mination. Hence  if  the  photometric  devices  employed  were 
correct,  the  values  should  agree  fairly  well,  and  their  uniformity 
is  a  test  of  the  method  rather  than  of  the  validity  of  the  law. 

The  subject  cannot  be  left  without  referring  to  the  psycho- 
physical  law  of  Fechner,  with  which  this  problem  is  intimately 
connected.  Briefly  stated,  Fechner's  law  is  that  as  a  stimulus 
increases  in  geometrical  progression  its  resulting  sensation 
increases  in  an  arithmetical  progression,  but  that  there  is  a 
slight  deviation  from  the  extreme  rigor  of  this  relation  when  the 
stimulus  becomes  very  intense  or  when  it  becomes  very  slight. 
The  relation  may  be  expressed  mathematically  by  the  formula 

S  =  C  log  R, 

where  S  is  the  intensity  of  the  sensation,  R  the  stimulus,  and 
C  a  constant.  In  this  particular  case  S,  which  is  the  sensation, 
is  equivalent  to  the  magnitude  of  the  star,  and  Ry  which  is  the 
stimulus,  is  equivalent  to  its  brightness,  but  as  we  have  no 
absolute  standard  for  brightness,  and  can  only  measure  it  rela- 
tively, we  must  compare  two  brightnesses. 

Let  A  and  B  be  the  brightnesses  of  two  stars,  and  Ml  and  M  2 
be  their  magnitudes.  Then 

Ml  =  C  log  A, 

M*=  C  log  B; 

hence  M l  —  Mz  =  C  (log  A  —  log  B), 

or  A  m  =  C  (log  A  —  log  B). 


STELLAR  MAGNITUDE  99 

This  may  be  written  also 

A  m 

—  =  log  A  -log  B, 

Am      A 

e^    =-. 

Let  e~c  =  p, 

and  we  have 

A-p*™ 
B~P     ' 

which  is  identical  with  equation  (2),  derived  before.  The  value 
of  p  may  be  found  as  described  above. 

It  should  be  stated  that  the  derivation  of  the  preceding  for- 
mula and  the  determination  of  the  value  of  p  by  Pogson  had 
been  published  before  Fechner  promulgated  his  law,  which  was 
not  until  1859;  and  the  latter  himself  states  that  this  very  rela- 
tion did  much  to  suggest  to  him  his  law,  and  he  considered  it 
the  most  important  confirmation  of  it.  It  has  just  been  stated 
that  Fechner's  law  is  not  considered  rigorous  for  the  extreme 
values  of  the  stimulus,  and  in  support  of  this,  evidence  has  been 
offered  from  the  study  of  stellar  magnitudes  showing  that  this 
ratio  is  not  absolutely  constant  throughout  the  scale.  It  seems 
hardly  consistent,  however,  to  the  author,  to  base  any  such 
reasoning  on  the  study  of  stellar  magnitudes,  since  at  best  they 
are  referred  to  standards  which  have  been  determined  empir- 
ically, and  rest  ultimately  on  the  classification  of  Ptolemy. 
Furthermore,  it  is  not  clear  that  the  magnitude  is  a  sensation 
of  which  the  brightness  is  the  stimulus,  because  the  use  of  mag- 
nitudes seems  to  be  merely  a  convenient  way  of  classifying  the 
brightnesses,  which  in  themselves  are  sensations.  This  is  not 
the  place  in  which  to  enter  into  a  psycho-physical  discussion, 
and  enough  has  been  said  to  present  the  problem.  It  will  be 
better  to  proceed  to  illustrate  the  application  of  the  formula, 
which  is  extremely  useful. 


100          THE  STUDY  OF  VARIABLE  STARS 


Examples  of  Pogson's  Rule 

1.  If  star  A  is  twice  as  bright  as  star  B,  what  is  the  difference 
in  magnitude?  Using  eq.  (4)  we  have 

1     - 
°g  B      log  2      0.301 


The  difference  in  magnitude  is  .75. 

2.  If  star  A  is  one  hundred  times  as  bright  as  star  B,  what 
is  the  difference  in  magnitude  ? 

A 

°g  B     log  100       2 


The  difference  in  magnitude  is  5. 

3.  The  faintest  star  visible  in  the  Lick  telescope  is  of  the  17th 
magnitude.    Polaris  is  the  standard  2nd  mg.  star.  What  is 
their  relative  brightness?  Using  eq.  (3)  we  have 

A 

log  —  =  A  m  log  /?, 

A 

log  —  =  15  x  0.4  =  6.0, 
£> 

A 

—  =  1,000,000. 

£> 

Polaris  is  a  million  times  as  bright  as  a  17th  mg.  star. 

4.  The  magnitude  of  Sirius  is  —1.43.   Find  its  light  ratio  to 
Polaris.  In  this  case  Sirius  is  star  A  and  Polaris  star  B,  and 
A  m  is  3.43. 

log  4  =  3.43  X  0.4  =  1.372, 

£> 

4  =  23.55. 

£> 

Sirius  is  23.55  times  as  bright  as  Polaris. 


STELLAR  MAGNITUDE  101 

5.  The  variable  star  Algol  loses  1.1  mg.  in  going  from  maxi- 
mum to  minimum.  What  per  cent  of  its  total  light  is  lost? 

Let      A  =  light  of  Algol  at  maximum  =  total  light, 
B  =  light  of  Algol  at  minimum; 

ft 

then    —  =  ratio  of  the  brightness  at  minimum  to  the  total 

A 

light, 

~D 

and     1  —  —  =  amount  of  light  lost, 
A. 

Am  =  1.1. 

From  eq.  (3) 
A 

log  —  =  A  m  log  p  =  1.1  X  0.4  =  0.44, 
z> 

T> 

log  —  -  9.56, 


1  -~A  =  .64. 
A 

The  amount  of  light  lost  is  64  per  cent  of  the  total  light. 

6.  If  at  the  time  of  minimum  a  star  has  lost  one  third  of  its 
light,  what  was  the  change  in  magnitude? 


0.176 


The  change  in  magnitude  is  .44  mg. 

We  have  also  the  following  table  of  relations  which  are  of 
convenient  use  in  making  visual  comparisons  of  stars.  For 
example,  if  we  decide  that  star  A  is  twice  as  bright  as  star  B, 


102 


THE  STUDY  OF  VARIABLE  STARS 


then  the  difference  in  magnitude  in  the  sense  A  —  B  is  .44.  If 
B  is  one  third  as  bright  as  A  the  difference  in  the  sense  A  —  B 
is  1.2  mg.  Only  a  few  values  such  as  are  most  likely  to  be  of 
frequent  use  are  included.  Other  desired  values  may  easily  be 
found. 


A 
B 

A  TO 

A-B 

B 
A 

Am 
A-B 

1.5 

.44 

.75 

.31 

2.0 

.75 

.67 

.44 

2.5 

1.00 

.50 

.75 

3.0 

1.19 

.33 

1.20 

4.0 

1.50 

.25 

1.50 

5.0 

1.75 

.10 

2.50 

CHAPTER  VI 
VISUAL  PHOTOMETRY 

VISUAL  observations  of  a  variable  star  consist  in  comparing 
its  brightness  with  that  of  another  star  which  is  supposed  to  be 
constant  in  brightness.  They  may  be  made  in  two  ways,  first 
by  direct  eye  estimates  of  the  comparative  brightness,  either 
with  or  without  a  telescope,  and  second  with  the  aid  of  a  pho- 
tometer attached  to  a  telescope.  Both  are  included  under  the 
heading  "Visual  Photometry,"  since  the  eye  itself  receives  the 
light  of  the  star. 

A  variable  star  should  be  compared  in  precisely  the  same 
manner  in  which  a  star  of  constant  brightness  would  be  com- 
pared, as  illustrated  by  the  work  of  Sir  William  Herschel;  that 
is,  by  placing  it  in  a  series  with  at  least  two  other  stars  and  indi- 
cating in  some  way  different  degrees  of  variation  in  its  bright- 
ness. The  method  most  commonly  used  is  that  of  Argelander, 
which  is  based  directly  on  that  of  Herschel,  its  chief  difference 
consisting  in  the  way  in  which  the  steps  are  indicated.  It  will 
be  remembered  that  Herschel  introduced  a  rather  complicated 
system  of  dots,  dashes,  and  commas,  to  represent  successive 
degrees  of  difference  in  brightness.  In  place  of  these  Argelander 
uses  ordinary  numbers,  and  the  varying  degrees  of  brightness 
he  calls  "steps,"  so  that  his  method,  which  is  now  in  very  gen- 
eral use,  is  called  Argelander's  step  method.  Though  the  main 
points  of  it  have  been  described  in  the  preceding  chapter,  in 
connection  with  the  work  of  Herschel,  a  repetition  of  them  in 
this  place  is  desirable  in  order  to  make  the  subject  complete. 
It  should  be  stated  that  the  form  in  use  was  of  gradual  develop- 
ment, and  that  at  times  Argelander  also  used  symbols  to  repre- 
sent steps.  These,  however,  are  not  recommended  to  the 
ordinary  observer,  and  hence  are  not  included  in  this  descrip- 


104          THE  STUDY  OF  VARIABLE  STARS 

tion.  A  full  account  of  them  will  be  found  in  Hagen's  Die 
Verdnderlichen  Sterne.  The  method  has  been  frequently  de- 
scribed, since  it  is  a  simple  way  of  making  comparisons  when 
no  photometer  is  at  hand.  Articles  on  variable  stars  and  hints 
for  making  observations  have  appeared  several  times  in  the 
Popular  Astronomy ,  to  which  the  reader  is  referred  for  further 
descriptions.1 

In  selecting  the  stars  with  which  to  compare  a  variable, 
those  nearest  to  it  in  brightness  should  be  chosen.  Two  at  least 
are  necessary,  one  a  little  brighter  and  one  a  little  fainter.  The 
variable  and  the  one  with  which  it  is  to  be  compared  should  be 
in  the  field  of  the  telescope  at  the  same  time  if  possible,  or  at 
least  should  not  be  so  far  apart  as  to  require  much  motion  of 
the  telescope  to  bring  them  into  the  field  in  rapid  succession. 
Some  recommend  that  the  observer  should  fix  his  eye  on  one 
star  at  a  time,  looking  at  it  intently  before  passing  to  the  other. 
There  are  such  frequent  fluctuations  in  the  atmosphere,  that  it 
is  only  by  this  method  that  he  can  obtain  a  distinct  impression 
of  its  brightness.  Alternate  observations  of  the  two  stars  will 
give  a  truer  impression  than  a  simultaneous  observation  of 
both.  In  making  the  comparison,  either  each  star  must  be 
brought  into  the  middle  of  the  field  by  moving  the  telescope, 
or  else  they  should  be  placed  symmetrically  with  reference 
to  the  middle  of  the  field.  In  contrast  to  the  above  rule,  the 
writer  has  often  found  it  very  useful  when  observing  faint  stars 
to  look  at  both  at  the  same  time,  using  averted  or  side  vision, 
for  the  following  reason.  It  is  known  to  every  observer  with 
the  telescope,  that  a  faint  object  at  the  limit  of  vision  can  be 
seen  more  easily  if  we  do  not  look  at  it  directly,  but  to  one  side, 
or  above  it,  or  if  the  eye  is  passed  rapidly  over  it  with  a  sweep- 
ing glance.  If  this  is  true  for  one  faint  star  it  is  equally  neces- 
sary when  comparing  two  which  are  at  the  limit  of  vision.  In 
observing  a  red  star  it  is  necessary,  on  the  other  hand,  to  take  a 

1  J.  A.  Parkhurst,  Pop.  Ast.,  I. 
P.  S.  Yendell,  Pop.  Ast.,  13,  14. 
W.  T.  Olcott,  Pop.  Ast.,  19. 


VISUAL  PHOTOMETRY  105 

rather  prolonged  steady  look  at  it,  since  the  impression  of  its 
brightness  deepens  as  one  looks. 

Having  selected  the  comparison  star  as  directed  above,  and 
having  given  the  steady  gaze,  the  observer  is  ready  to  make  his 
estimate.  Since  the  stars  are  always  fluctuating  in  brightness, 
one  glance  is  not  sufficient  to  enable  the  observer  to  make  a 
definite  decision,  hence  he  should  look  several  times,  each  time 
making  an  estimate  and  in  the  end  adopting  that  one  which 
occurs  most  frequently.  As  Herschel  suggests,  when  there  is 
very  great  difficulty  in  coming  to  a  decision,  one  should  cast  the 
eyes  upon  the  ground,  and  when  they  are  refreshed  look  again 
upon  the  stars  to  be  compared.  This  can  be  done  literally  only 
when  observing  in  the  open  without  the  telescope,  but  the 
principle  is  the  same,  and  needs  no  further  elucidation.  When 
the  eye  is  tired  and  the  mind  is  fatigued  with  its  own  indecision, 
a  brief  rest  will  bring  a  quicker  decision  in  its  train. 

If  as  a  result  of  such  careful  comparison,  the  two  stars,  which 
may  be  called  a  and  #,  seem  to  be  of  absolutely  equal  bright- 
ness, they  should  be  written  down  a  0  v,  or  simply  a  v,  or  a  =  v. 
Also  if  they  do  not  seem  to  be  equal,  but  half  the  time  a  appears 
to  be  a  very  little  brighter,  and  half  the  time  v  is  the  brighter, 
still  the  final  result  will  be  a  0  v.  This  Herschel  represented  by 
the  period,  "  ."  ,  as  a  symbol. 

If  it  happens,  on  the  other  hand,  that  star  a  appears  the 
brighter  of  the  two  rather  more  than  half  the  time,  but  that 
there  is  an  element  of  doubt  in  the  observer's  mind  as  to  which 
is  really  the  brighter,  while  sometimes  a  appears  equal  to  v, 
then  we  can  say  that  a  is  one  step  brighter  than  v,  or  a  1  v,  the 
brighter  star  being  written  first.  If  one  star  is  certainly  brighter 
than  the  other  and  yet  only  very  little  brighter,  the  interval  is 
called  two  steps  and  written  a  2  v.  Sometimes  this  difference  is 
expressed  by  saying  that  one  star,  and  only  one,  can  be  inter- 
mediate between  them  in  brightness.  Three  grades  represent  a 
difference  about  which  there  is  no  doubt,  but  still  not  a  great 
one.  The  observer  can  have  no  difficulty  in  defining  it  for  him- 
self. Higher  than  this  it  is  not  advisable  to  go  by  the  simple 


106          THE  STUDY  OF  VARIABLE  STARS 

step  method,  as  the  uncertainty  is  too  great.  There  are  other 
devices,  however,  by  means  of  which  larger  differences  can  be 
estimated  with  considerable  degree  of  accuracy  when  necessity 
arises,  as  it  often  does  when  the  variable  is  bright  and  compari- 
son stars  are  not  numerous.  One  of  these  devices  is  to  decide 
the  relative  brightness  between  the  two  stars;  as  for  instance, 
a  may  appear  to  be  twice  as  bright  as  v,  meaning  that  if  another 
star  as  bright  as  the  variable  were  to  be  added  to  it,  the  result 
would  be  as  bright  as  a.  From  the  table  in  the  preceding  chap- 
ter, built  upon  Pogson's  rule,  this  will  correspond  to  a  differ- 
ence in  magnitude  of  .75.  Or  if  it  is  decided  that  v  is  one  third 
as  bright  as  a,  then  a  is  1.1  mg.  brighter.  When  estimates  are 
made  in  this  manner,  the  greatest  care  is  necessary  in  writing 
them  down.  In  the  first  case  it  would  be  quite  incorrect  to 
write  a  2  v,  since  this  signifies  a  difference  of  only  two  steps, 
which  is  a  very  much  smaller  difference;  but  the  full  expression, 
a  is  twice  as  bright  as  v,  or  v  is  one  third  as  bright  as  a,  should  be 
used.  The  writer  has  frequently  employed  this  method  with 
satisfactory  results. 

In  making  comparisons  for  the  brightness  of  a  variable,  more 
than  one  star  should  be  used.  The  best  combination  is  to  take 
two,  one  a  little  brighter  and  one  a  little  fainter.  If  convenient 
a  third  should  be  added,  so  as  to  have  as  many  independent 
determinations  of  the  magnitude  as  possible. 

The  use  of  two  comparison  stars,  one  brighter  and  one  fainter, 
makes  it  possible  to  employ  still  another  method  of  comparison 
which  was  first  recommended  by  Pickering.  The  two  stars 
selected  should  not  differ  more  than  a  magnitude.  Let  them 
be  called  a  and  b.  Estimate  the  brightness  of  v  in  tenths  of  the 
interval  from  a  to  b.  Thus  if  v  is  half  way  between  a  and  b  in 
brightness,  call  it  a  5  v  5  b,  or  omitting  v  write  a  5  b.  If  a  is  not 
much  brighter  than  v,  we  may  write  a  1  b  or  a  %  b,  and  if  the 
variable  is  nearer  b  in  brightness,  we  should  write  a  8  b  or  a  9  b. 
This  method,  while  advantageous  when  it  is  necessary  to 
employ  stars  which  differ  widely  in  brightness,  is  not  in  general 
use.  The  writer  has  occasionally  used  a  modification  of  it  as 


VISUAL  PHOTOMETRY  107 

follows:  instead  of  dividing  the  interval  in  brightness  between 
a  and  6  into  ten  parts,  any  convenient  unit  may  be  used,  for 
example,  if  the  variable  seems  to  be  two  fifths  of  the  difference 
in  brightness  from  a  it  could  be  written  a%v3b.  a3v4sb  would 
show  that  v  is  three  sevenths  of  the  difference  fainter  than  a. 
Variations  are  thus  quite  permissible,  so  long  as  the  method  of 
notation  is  fully  understood. 

Another  method,  that  of  estimating  the  magnitude  directly, 
also  devised  by  Pickering,  is  used  at  present  at  the  Harvard 
Observatory,  and  recommended  by  him  to  other  astronomers. 
In  order  to  employ  it,  however,  the  maps  must  be  specially 
prepared.   This  is  done  by  attaching  to  each  comparison  star 
its  magnitude,  omitting  the  decimal  point  to  avoid  confusion 
with  the  faint  stars.  The  magnitudes  are  taken  from  the  photo- 
metric measurements  especially  carried  on  for  the  purpose,  the 
results  of  which  are  published  in  the  various  H.C.O.  Annals. 
These  charts  may  be  had  by  any  observer  from  the  Harvard 
Observatory.  This  method  has  been  found  very  successful  and 
expeditious  at  Cambridge.  The  chief  criticism  that  can  be  made 
is  that  there  is  no  possibility  of  discovering  a  mistake  by  going 
back  to  the  original  comparison  to  see  if  a  wrong  star  has  been 
used.   It  has  also  been  objected  that  if  there  should  at  some 
time  be  a  change  in  the  magnitudes  assigned  to  the  comparison 
stars,  it  would  not  be  possible  to  correct  the  observations,  since 
it  would  not  be  known  which  stars  had  been  used.   To  this 
Pickering  replied  that  if  they  were  changed,  a  curve  could  be 
drawn  which  would  show  the  differences  between  the  two  sys- 
tems of  magnitudes,  and  corrections  could  be  applied  to  the 
resulting  magnitudes  of  the  variable.   The  writer  would  ven- 
ture to  say  by  way  of  comment,  that  while  the  above  method  is 
very  suitable  to  observers  on  the  staff  in  Cambridge,  it  is  not 
so  satisfactory  to  the  ordinary  observer,  who  may  wish  to  find 
the  steps  of  his  comparison  stars,  or  who  wishes  to  have  them 
in  his  record. 

Various  modifications  of  Argelander's  step  method  have  been 
introduced  by  different  observers.   A  full  treatment  may  be 


108          THE  STUDY  OF  VARIABLE  STARS 

found  in  Hagen's  Die  Verdnderlichen  Sterne,  chap.  x.  It  has 
not  been  thought  necessary  to  give  them  here. 

Mention  may  be  appropriately  made  at  this  point  of  some  of 
the  precautions  which  must  be  observed  in  making  these  com- 
parisons. Some  of  these  precautions  relate  to  the  color  of  the 
star,  and  others  have  to  do  with  the  instrument  and  atmos- 
pheric conditions. 

Connected  with  color  we  have  two  well-known  phenomena 
which  are  called  respectively  the  Purkinje  phenomenon  and 
that  of  Dove.  A  simple  way  of  stating  the  former  is  as  follows: 
if  the  observer  starts  with  two  equal  red  and  green  lights  and 
increases  them  both  in  brightness  in  the  same  degree,  the  red 
light  will  appear  brighter  than  the  green  one,  and  if  they  are 
diminished,  the  red  will  grow  faint  more  rapidly,  that  is,  the 
red  light  changes  more  rapidly,  both  in  increasing  and  in  dimin- 
ishing. This  is  easily  illustrated  by  watching  articles  in  a  room 
as  it  is  growing  dusk.  If  there  are  two  books  in  a  case,  one  bound 
in  red  and  the  other  in  green,  appearing  equally  conspicuous  in 
ordinary  daylight,  as  the  room  grows  darker  the  green  one  will 
remain  visible  longer  than  the  red  one;  and  conversely,  if  at  the 
other  end  of  the  day,  an  observer  is  minded  to  watch  them  in 
the  early  light  of  dawn,  the  red  will  be  seen  to  grow  bright  more 
rapidly.  This  fact  bears  directly  upon  the  observation  of  long 
period  variables,  since  many  of  them  are  quite  reddish  in  color. 
As  one  of  these  approaches  maximum,  i.e.,  grows  brighter,  it 
will  have  a  different  rate  of  increase  from  a  white  star,  and  will 
appear  brighter  than  a  white  star  which  increases  intrinsically 
at  the  same  rate  and  to  the  same  amount.  However,  there  is  no 
way  in  which  this  difficulty  can  be  overcome.  All  that  can  be 
done  is  to  select  a  red  comparison  star  whenever  that  is  possi- 
ble, and  further  than  this  to  make  comparisons  as  carefully  as 
possible,  taking  a  long  look  at  the  reddish  star,  as  by  so  doing 
its  brightness  becomes  more  vivid.  Care  should  also  be  used  in 
combining  observations  made  with  different  telescopes.  If  two 
stars  of  which  one  is  red  are  observed  together,  first  with  one 
telescope,  and  then  with  one  of  larger  aperture,  the  Purkinje 


VISUAL  PHOTOMETRY  109 

phenomenon  is  exactly  reproduced,  because  we  have  objects  of 
different  colors  viewed  first  with  one  degree,  and  afterward 
with  another  greater  degree,  of  illumination.  The  relative 
difference  will  be  greater  in  the  first  case  than  in  the  second, 
therefore  in  the  case  of  a  red  star,  the  same  telescope  should  be 
used  throughout  the  observations.  Attention  is  called  to  the 
fact  that  while  stars  are  never  green,  and  rarely  bluish,  the  dif- 
ference between  a  red  star  and  a  white  one  is  of  the  same  order 
as  that  between  a  red  and  a  green  one. 

Dove  discovered  that  the  brilliancy  of  the  background  has  an 
important  effect  upon  the  relative  brightness  of  different  colors. 
When  the  background  is  very  bright,  the  red  star  is  the  brighter, 
and  when  it  is  relatively  faint  or  dark,  the  bluish  object  appears 
brighter.  This  has  an  astronomical  bearing,  since  observations 
made  during  bright  moonlight  or  twilight  will  present  colors 
against  a  brilliant  background,  and  hence  make  a  red  star  the 
brighter. 

Other  precautions  depend  upon  the  position  of  the  stars  in 
the  field  of  view  of  the  telescope.  A  star  near  the  edge  of  the 
field  appears  brighter  than  one  in  the  middle,  therefore,  when 
the  two  to  be  compared  are  not  far  apart,  they  should  be  placed 
at  equal  distances  from  the  center  of  the  field.  When  this  is  not 
possible,  each  star  should  in  turn  be  brought  into  the  center  of 
the  field  and  looked  at  steadily  until  a  distinct  impression  of  its 
brightness  is  obtained. 

Another  error  not  so  easily  avoided  arises  from  what  is  called 
the  parallactic  angle,  meaning  the  angle  which  the  line  joining 
the  two  stars  makes  with  the  vertical  to  the  horizon.  A  star 
which  is  lower  in  the  field  usually  appears  brighter,  and  some 
observers,  for  instance,  Yendell,  make  the  difference  as  great  as 
half  a  magnitude.  The  reason  is  fatigue  of  the  retina,  and  is 
explained  as  follows.  The  image  on  the  retina  is  inverted.  Dur- 
ing the  daytime,  the  light  from  the  sky  falling  upon  the  lower 
part  of  the  retina  fatigues  it,  as  does  also  most  artificial  light  at 
night,  since  electric  and  gas  lights  are  in  general  suspended 
from  the  ceiling.  Darker  objects  such  as  furnishings,  trees,  etc., 


110          THE  STUDY  OF  VARIABLE  STARS 

form  their  images  on  the  upper  part  of  the  retina  and  do  not 
fatigue  it  as  much.  Hence  when  the  eye  is  used  for  observation 
at  night,  the  light  from  the  star  in  the  lower  part  of  the  field 
falls  upon  that  part  of  the  retina  which  is  least  fatigued,  and 
hence  appears  brighter  than  a  star  in  the  upper  part  of  the  field. 
That  the  physiological  effect  varies  with  the  observer  is  doubt- 
less true,  since  there  is  a  difference  in  the  native  strength  of  the 
eye,  and  also  in  the  daily  surroundings  depending  upon  the 
ordinary  occupation,  which  may  tend  to  obviate  this  difficulty. 
At  any  rate  some  observers  do  not  find  it  so  marked  as  others. 
The  best  way  to  surmount  it  is  to  turn  the  head  so  that  the  line 
joining  the  eyes  is  parallel  to  the  line  joining  the  stars.  The 
same  effect  is  produced  by  using  a  reversing  prism  which  may 
be  adjusted  so  as  to  change  the  position  of  the  line  joining  the 
stars. 

Other  precautions  are  of  a  different  character.  Flying  clouds 
and  moonlight  are  to  be  avoided,  especially  the  former.  Doubt- 
ful observations  should  be  rejected  at  once.  The  eye  should 
have  complete  rest  in  a  darkened  room  for  about  ten  minutes 
before  beginning  to  observe,  at  least  after  having  been  used  in 
a  brightly  lighted  room  for  reading.  To  illuminate  the  paper  for 
making  the  record,  some  recommend  the  use  of  a  lantern  cov- 
ered with  red,  which  is  just  bright  enough  to  allow  the  observer 
to  study  the  star  maps  and  to  see  to  write. 

Other  causes  of  error  to  be  avoided  will  be  mentioned  in  the 
chapter  entitled  "Hints  for  Observers." 

We  pass  now  to  the  description  of  several  different  kinds  of 
photometric  apparatus  which  have  been  devised  to  assist  the 
observer  hi  making  visual  comparisons.  In  order  fully  to  under- 
stand their  construction  it  will  be  necessary  to  consider  some  of 
the  underlying  principles  of  physics  upon  which  they  are  based. 
In  the  introductory  chapter  there  was  given  a  statement  of  the 
principle  of  refraction,  on  which  the  construction  of  the  spectro- 
scope is  based,  and  also  an  account  of  the  wave  theory  of  light. 
It  was  there  stated  that  a  ray  of  light,  in  passing  from  a  medium 
of  one  density  to  that  of  another,  is  bent  with  reference  to  the 


VISUAL  PHOTOMETRY  ill 

normal  at  the  point  of  incidence.  In  passing  from  a  rarer  to  a 
denser  medium  it  is  bent  toward  the  normal,  and  in  passing 
from  a  denser  to  a  rarer  medium  it  is  bent  from  the  normal. 
The  law  according  to  which  this  deviation  takes  place  was  dis- 
covered in  1621  by  Snell,  and  is  expressed  by  the  following 
formula:  — 

sin  i 

~~.       =  n, 
sin  r 

that  is,  the  ratio  between  the  sine  of  the  angle  of  incidence  and 
the  sine  of  the  angle  of  refraction  is  constant  for  any  two  given 
media.  When  air  is  the  rarer  medium  n  is  called  the  index  of 
refraction.  This  law  was  later  found  to  be  not  quite  rigorous, 
for  the  index  of  refraction  varies  slightly  with  the  tempera- 
ture, barometric  pressure,  and  wave-length  of  the  incident  ray. 
Hence  when  the  value  of  n  is  given  it  is  understood  that  stand- 
ard conditions,  i.e.,  760  mm.  barometric  pressure,  0°  C.  temper- 
ature, were  employed,  and  that  white  light  was  used.  This 
bending  of  the  ray  of  light  usually  has  no  effect  upon  the  wave 
motion  which  causes  it.  When  the  incident  beam  falls  upon  an 
ordinary  transparent  substance  its  ether  particles  are  vibrating 
in  all  directions  at  right  angles  to  that  in  which  the  wave  mo- 
tion is  being  propagated.  After  it  emerges  and  the  direction  of 
propagation  has  been  altered,  the  ether  vibrations  are  still 
perpendicular  in  all  directions  to  it. 

More  careful  examination  shows  that  there  are  some  sub- 
stances which  do  affect  the  vibrations  in  the  light  waves,  so 
that  when  they  emerge  they  are  no  longer  transverse  in  all 
directions,  but  are  confined  to  two,  which  are  at  right  angles  to 
each  other.  One  of  these  substances  is  a  colorless  crystal  called 
Iceland  spar,  which  is  a  natural  crystal,  bounded  by  six  plane 
faces,  lying  parallel,  two  and  two.  If  a  beam  falls  upon  one  of 
its  faces  at  right  angles,  two  beams  are  seen  to  emerge  from  the 
opposite  face,  which  are  equally  bright,  but  each  of  which  has 
half  the  brightness  of  the  incident  beam.  One  of  these  follows 
the  direction  of  the  normal,  thus  obeying  the  ordinary  law  of 
refraction,  while  the  other  is  displaced  to  one  side.  This  phe- 


112          THE  STUDY  OF  VARIABLE  STARS 

nomenon  is  known  as  double  refraction,  and  produces  very 
interesting  results  when  the  crystal  is  rotated.  They  may  be 
studied  by  placing  the  crystal  on  a  table  over  a  piece  of  white 
paper  on  which  is  a  black  pencil  dot;  or  better  still,  by  allowing 
a  ray  of  sunlight,  which  enters  through  a  small  round  aperture, 
to  pass  through  the  crystal  and  fall  upon  a  screen  suitably 
placed.  The  crystal  should  then  be  rotated  in  the  plane  in 
which  it  is  set,  i.e.,  if  placed  on  a  table,  it  should  be  kept  flat 
during  rotation.  When  this  is  done,  the  beam  which  emerges 
in  the  direction  of  the  normal  remains  unchanged  in  position, 
while  the  other  circles  about  it.  Thus  the  one  ray  behaves  in  a 
perfectly  normal  manner,  while  the  other  is  refracted  differ- 
ently, and  its  displacement  seems  to  be  connected  with  the 
position  of  the  crystal.  If  the  crystal  is  inclined  to  the  incident 
beam  the  one  ray  still  continues  to  obey  SnelPs  law  in  the  ordi- 
nary manner,  and  hence  is  called  the  ordinary  ray,  while  the 
other  is  refracted  at  angles  which  differ  according  to  the  angle 
of  incidence,  but  do  not  obey  the  law.  Since  this  ray  behaves  in 
this  unusual  manner  it  is  called  the  extraordinary  ray.  The  index 
of  refraction  of  the  ordinary  ray  is  found  by  the  usual  manner, 
while  that  of  the  extraordinary  ray  is  more  complicated,  and 
depends  upon  the  path  of  the  rays  with  reference  to  the  form 
of  the  crystal. 

It  is  not  necessary  for  our  purpose  to  investigate  this  matter 
further.  We  may  pass  on  to  the  phenomena  which  take  place 
when  the  beam  of  light  passes  through  a  second  crystal  of  spar, 
an  experiment  which  was  first  performed  by  Huyghens.  He 
found  that  each  of  the  two  rays  emerging  from  the  first  crystal, 
after  passing  through  the  second,  was  divided  into  two  others, 
making  four  in  all;  but  that  the  four  were  of  unequal  brilliancy, 
the  relative  brightness  depending  upon  the  position  of  the  sec- 
ond crystal.  Before  describing  this  effect  more  in  detail  it  is 
necessary  to  mention  some  further  facts  in  regard  to  the  crystal 
itself.  Iceland  spar  crystallizes  in  different  forms,  but  can 
readily  be  split  into  small  crystals  of  a  certain  definite  shape, 
breaking  along  certain  planes  which  are  called  planes  of  cleav- 


VISUAL  PHOTOMETRY  113 

age.  The  bounding  surfaces  are  six  parallelograms,  equal  two 
and  two,  each  one  of  which  has  for  its  face  angles  two  acute  and 
two  obtuse  angles.  There  are  eight  solid  angles,  two  of  which 
are  formed  by  the  junction  of  three  obtuse  angles.  A  line  which 
passes  through  the  vertex  of  one  of  these  angles  and  is  equally 
inclined  to  the  three  adjoining  faces  is  called  the  axis  of  the 
crystal.  This  axis  has  special  optical  properties,  which  are  not 
altered  by  the  dimensions  of  the  crystal,  nor  reduced  by  further 
cleavage,  and  may  be  considered  simply  as  giving  a  direction  in 
the  crystal.  We  are  now  ready  to  study  the  effect  of  passing  a 
single  beam  of  light  through  two  crystals  of  Iceland  spar. 

Suppose  that  the  beam  of  light  falls  perpendicularly  upon 
the  first  crystal,  and  that  the  second  crystal  is  placed  so  that  its 
axis  is  parallel  to  that  of  the  first,  in  which  case  the  two  crystals 
are  said  to  be  parallel.  On  the  screen  will  appear  two  spots  of 
light,  of  equal  brightness,  and  nearly  equal  to  those  emerging 
from  the  first  crystal,  the  diminution  being  due  only  to  absorp- 
tion on  passing  through  the  crystals.  Now  rotate  the  second 
crystal  ever  so  slightly.  Immediately,  two  new  spots  of  light 
which  are  quite  faint  appear,  while  those  already  present  grow 
less  bright.  As  the  rotation  is  continued  the  two  faint  ones  grow 
brighter,  and  the  bright  ones  fainter,  and  by  the  time  the  rotat- 
ing crystal  has  been  turned  through  an  angle  of  about  45°  the 
four  images  are  practically  equal.  At  90°,  when  the  two  crystals 
are  said  to  be  crossed,  the  images  which  appeared  first  have 
become  extinguished,  and  the  new  ones  have  reached  their  max- 
imum brightness.  The  four  are  again  equal  at  135°;  and  at  180° 
the  original  ones  have  reached  their  maximum  brightness,  as 
the  crystals  are  again  parallel,  and  so  on.  That  is,  there  are 
four  positions  in  which  the  images  are  equal,  and  four  positions 
in  which  only  two  appear,  but  in  the  latter  case  they  are  in  two 
sets,  one  pair  being  identical  at  0°  and  180°,  and  the  second 
pair  at  90°  and  270°.  The  motions  of  the  four  spots  of  light  are 
also  quite  complicated,  but  as  they  have  no  particular  bearing 
upon  the  problem  in  hand,  which  is  one  of  measuring  the 
brightness  of  a  star,  they  require  no  further  description  at  this 


114          THE  STUDY  OF  VARIABLE  STARS 

point.  The  important  fact  is  that  a  method  has  been  found  by 
which  the  observer  can  diminish  at  will  the  brightness  of  a  light, 
until  it  is  equal  to  another  light  with  which  it  is  compared,  or 
until  it  is  extinguished;  in  addition,  the  number  of  degrees 
through  which  the  crystal  is  turned  in  order  to  produce  the 
desired  effect  can  be  measured  exactly  and  made  the  basis  of 
further  calculations.  The  practical  application  of  this  to  photo- 
metric apparatus  will  be  described  presently,  as  it  is  necessary 
first  to  give  some  theoretical  explanation  of  the  phenomenon  of 
double  refraction. 

It  is  evident  that  the  crystal  of  Iceland  spar  has  a  special 
effect  upon  the  rays  of  light  passing  through  it,  and  when  they 
emerge  from  it  they  will  not  pass  through  a  second  crystal 
freely,  but  are  affected  by  the  relative  positions  of  the  two.  This 
is  readily  explained  by  a  reference  to  the  theory  of  light  waves. 
The  structure  of  the  crystal  is  such  that  it  will  not  permit 
transverse  waves  to  pass  through  it  in  all  directions,  but  only 
in  two,  which  are  at  right  angles  to  each  other.  All  of  the 
vibrations  in  the  waves  forming  the  ordinary  beam  are  in  one 
direction  and  all  of  those  in  the  extraordinary  beam  are  at  right 
angles  to  it.  There  is  nothing  in  the  appearance  of  either  ray 
to  indicate  this,  and  it  is  only  discovered  when  they  fall  upon 
the  second  crystal,  and  are  split  up  into  components,  parallel 
to  the  axis  of  the  second  crystal  and  at  right  angles  to  it,  result- 
ing in  the  four  images.  When  light  has  been  altered  in  this  way 
it  is  said  to  be  polarized,  and  the  crystal  which  produces  the 
alteration  is  called  the  polarizer,  and  the  second  crystal,  which 
is  used  to  find  out  the  fact,  is  called  the  analyzer.  Many  sub- 
stances besides  Iceland  spar  produce  the  same  effect;  even 
reflection  from  glass,  or  from  the  clouds  at  a  certain  angle,  will 
produce  polarization.  The  fact  can  only  be  discovered  by  study- 
ing the  light  with  a  second  crystal  to  see  whether  by  rotating 
it  the  light  is  diminished  in  brightness. 

It  may  readily  be  inferred  that  this  principle  can  be  applied 
to  the  study  of  stellar  brightness.  If  the  light  from  a  star  can 
first  be  polarized,  then  passed  through  an  analyzer,  and  the 


VISUAL  PHOTOMETRY 


115 


analyzer  rotated,  the  light  from  the  star  will  be  diminished  and 
finally  become  extinguished.  If  it  is  compared  with  some  stand- 
ard light,  it  can  be  made  equal  to  it  under  certain  conditions 
which  are  under  the  control  of  the  observer  and  are  susceptible 
of  exact  measurement.  This  statement  is  intended  only  to  indi- 
cate how  the  principle  of  polarized  light  may 
be  applied  to  photometric  observations,  but 
the  simple  crystal  cannot  be  used,  with  its 
multiplicity  of  images,  and  some  device  must 
be  introduced  which  will  reduce  them.  This 
might  possibly  be  accomplished  by  the  use 
of  a  screen  to  stop  out  one  of  the  rays  from 
the  first  crystal,  but  the  angle  separating 
them  is  not  large,  and  hence  the  incident 
beam  would  have  to  be  of  very  small  diam- 
eter, or  else  the  piece  of  spar  very  thick. 
Instead  a  convenient  contrivance  is  used 
which  is  called  a  "Nicol's  prism."  It  is  made 
by  splitting  an  ordinary  crystal  by  a  plane 
which  passes  through  the  axis,  or  along  the 
line  ABy  as  is  illustrated  in  the  accompanying 
figure,  and  is  perpendicular  to  the  principal 
plane  for  the  face  AC. 

The  two  cut  faces  are  then  polished  smooth 
and  fastened  together  again  in  their  original 
position  with  Canada  balsam.  This  has  an 
index  of  refraction  such  that  by  the  well- 
known  principle  of  total  reflection  the  ordi- 
nary ray  is  turned  aside  at  the  joining  surfaces 
and  reflected  out,  so  that  only  the  extraor- 
dinary ray  passes  through  such  a  prism,  which  then  acts  as  a 
polarizer.  If  a  second  NicoFs  prism,  which  also  permits  only 
the  extraordinary  ray  to  pass  through,  is  placed  in  the  path  of 
this  beam  and  is  rotated,  the  emerging  beam  will  diminish  to 
extinction,  then  increase  to  maximum,  and  so  on,  being  maxi- 
mum twice  and  zero  twice.  The  amount  of  rotation  can  be 


THE  NICOL'S 
PRISM 


116          THE  STUDY  OF  VARIABLE  STARS 

measured  by  a  graduated  circle  and  the  result  used  in  the  de- 
terminations of  brightness  in  accordance  with  the  law  known 
as  that  of  Malus.  "  If  a  beam  of  light  already  polarized  falls 
upon  the  doubly  refracting  crystal  the  intensities  of  the  two 
emergent  beams,  the  ordinary  and  extraordinary,  are  propor- 
tional respectively  to  the  cosine 2  and  sine 2  of  the  angle  which 
the  plane  of  polarization  of  the  incident  beam  makes  with  the 
plane  of  the  principal  section  of  the  crystal."1 

If  a  beam  of  light  passes  through  two  Nicols,  and  they  are  so 
placed  that  their  principal  sections  are  parallel,  then  the  plane 
of  polarization  of  the  extraordinary  ray  coming  from  the  first 
Nicol  will  be  perpendicular  to  the  principal  section  of  the  sec- 
ond Nicol,  and  hence  the  intensity  of  the  emergent  beam  will  be 

sin2  90°  =  1; 

that  is,  a  maximum.  If  we  rotate  the  Nicols  with  reference  to 
each  other,  then  the  intensity  will  diminish  in  proportion  to 
the  square  of  the  sine  of  the  angle  of  rotation.  The  practical 
working  of  this  principle  will  be  more  readily  understood  by  a 
description  of  a  photometer  which  is  based  upon  it.  The  one 
best  suited  to  this  purpose,  and  also  one  which  is  in  very  gen- 
eral use  in  Germany,  is  known  as  the  Zbllner  photometer,  which 
will  now  be  described.  While  its  construction  may  seem  some- 
what complicated,  it  is  understood  to  be  quite  convenient  in 
use. 

The  photometer  and  the  telescope  may  be  built  as  one  instru- 
ment, in  which  case  the  alt-azimuth  style  of  mounting  is  used; 
or  the  photometer  is  made  separately  in  such  a  way  that  it  can 
be  attached  to  an  ordinary  telescope.  Figure  18  shows  the  form 
in  common  use,  and  is  excellent  for  indicating  the  positions  of 
the  different  parts.  It  illustrates  very  well  the  general  device 
that  is  adopted  whenever  an  artificial  star  is  used  for  compar- 
ison. An  arm  is  attached  to  the  telescope  tube  at  right  angles 
to  it,  which  holds  the  apparatus  for  forming  the  artificial  star, 
as  well  as  that  used  to  diminish  its  brightness  and  make  it 
equal  to  the  real  star,  for  obviously  the  comparison  star  must 
1  H.  Kobold,  Der  Ban  des  Fixsternsgstems,  19. 


VISUAL  PHOTOMETRY 


117 


be  the  brighter  to  begin  with,  since  the  image  of  the  real  star 
is  formed  in  the  usual  way  by  light  coming  down  the  tube  of 
the  telescope,  and  cannot  be  altered.  Opposite  the  opening  of 


Figure  18 

THE  ZOLLNER  PHOTOMETER 

this  side  arm  is  placed  a  mirror  of  plane  glass  which  serves  to 
reflect  the  light  from  the  artificial  star  down  the  tube,  where 
an  image  is  formed  in  the  field  with  that  of  the  real  star;  the 
reducing  apparatus  is  then  adjusted  until  the  two  are  equal, 
and  by  a  proper  computation  the  magnitude  may  be  obtained. 


118         THE  STUDY  OF  VARIABLE  STARS 

The  preceding  statement  merely  outlines  the  construction  of 
the  instrument,  of  which  a  full  description  will  now  be  given. 
It  should  be  stated  also  at  this  point,  that  this  photometer 
has  an  additional  device  by  means  of  which  the  color  of  the 
artificial  star  can  be  altered  so  as  to  resemble  closely  that  of 
the  real  star,  a  very  important  object  to  be  attained. 

The  light  which  produces  the  artificial  star  is  from  a  petro- 
leum lamp  tightly  enclosed  so  that  no  light  can  escape  from  it 
except  through  a  small  orifice  just  opposite  the  end  of  the  arm 
CD.  Emerging  from  this  the  light  passes  through  a  pinhole 
diaphragm,  o,  in  the  end  of  the  side  tube.  In  later  forms  of  the 
instrument,  this  diaphragm  has  several  holes  of  different  sizes 
in  order  to  simulate  stars  of  varying  magnitudes.  The  rays 
diverge  as  they  come  through  this  opening,  and  are  made  to 
separate  still  more  by  passing  through  a  double  concave  lens, 
m,  which  has  the  effect  of  making  the  image  appear  quite  small. 
They  then  fall  upon  a  Nicol  prism,  k,  which  polarizes  the  light, 
and  if  passed  through  a  second  Nicol,  i,  by  rotation  can  be 
diminished  in  brightness  until  made  equal  to  the  real  star.  But 
at  this  point  the  device  is  introduced  by  which  the  color  is 
altered.  It  consists  of  a  thin  plate  of  quartz,  Z,  cut  perpendicu- 
lar to  the  axis  of  the  crystal.  This  is  placed  next  to  the  Nicol 
just  mentioned,  and  is  closely  followed  by  another  Nicol,  i. 
The  second  Nicol  and  the  quartz  section  are  fixed  with  refer- 
ence to  each  other,  but  the  first  Nicol  can  be  rotated,  and 
through  its  rotation,  by  the  principles  of  interference,  the  color 
is  changed  to  agree  with  that  of  the  real  star,  and  in  this  posi- 
tion all  three  pieces  can  be  clamped  together.  The  light  as  it 
emerges  from  the  second  Nicol  is  thus  polarized  and  colored. 
It  then  passes  through  a  third  Nicol,  h,  which  acts  as  an  ana- 
lyzer, and  on  rotation  diminishes  it  until  it  is  equal  in  bright- 
ness to  the  real  star.  In  practice,  however,  this  third  prism  is 
stationary,  and  the  system  which  contains  the  quartz  is  rotated, 
the  angle  being  read  from  the  scale  nn' '.  The  beam  of  light  then 
passes  through  a  double  convex  lens,  /,  which  serves  to  collect 
the  rays  and  throw  them  upon  the  plane  glass  mirror,  ee', 


VISUAL  PHOTOMETRY  119 

which  reflects  them  down  the  tube  of  the  telescope,  forming  an 
image  at  the  focal  point  side  by  side  with  that  of  the  real  star,  b. 
Since  this  mirror  is  thick  enough  to  reflect  from  both  front  and 
back  surfaces,  there  will  be  two  images  of  the  artificial  star,  gg, 
one  on  each  side  of  the  true  one.  The  order  of  adjustment  of 
the  different  parts  of  the  instrument  is  as  follows:  the  two 
Nicols  i  and  k  are  set  parallel,  the  telescope  is  pointed  so  that 
the  star  to  be  observed  is  in  the  field,  the  first  Nicol  k  is  turned 
until  the  light  of  the  artificial  star  has  the  same  color  as  the 
real  star,  after  which  the  two  prisms  k  and  i  and  the  quartz 
crystal  I  are  clamped  together.  This  part  is  then  turned  until 
the  artificial  star  has  become  diminished  in  brightness  into 
equality  with  the  true  star,  and  the  circle  by  which  it  is  turned 
is  read  in  degrees.  The  rotation  is  continued  until  equality  is 
again  secured  and  another  reading  taken,  there  being  four 
positions  of  the  Nicol  in  which  this  will  occur.  From  the  four 
readings  the  brightness  of  the  real  star  can  be  determined  in 
comparison  with  that  of  the  artificial  star  taken  as  a  stand- 
ard. 

The  difference  in  magnitude  can  be  obtained  by  combining 
the  results  in  accordance  with  the  law  of  Malus  and  Pogson's 
rule,  as  the  following  example  will  show. 

If  7  is  the  angle  through  which  the  Nicol  is  turned  in  order 
to  produce  equality  of  light,  then  the  intensity  of  the  image 
formed  would  be  measured  by  X  sin2  1,  where  X  is  the  inten- 
sity of  the  incident  beam;  that  is,  the  real  star  A  has  a  bright- 
ness X  sin2  I,  where  X  is  the  brightness  of  the  artificial  star. 
The  brightness  of  the  standard  star,  B,  must  be  measured  in 
the  same  way,  and  will  be  represented  by  X  sin2  /',  where  I' 
is  the  angle  through  which  the  Nicol  is  turned  in  order  to  pro- 
duce equality  of  light  for  the  standard  star.  Hence  the  ratio 
in  brightness  will  be 

A     sin2  1 


This  may  be  reduced  to  a  difference  in  magnitude  by  Pogson's 
rule;  — 


120          THE  STUDY  OF  VARIABLE  STARS 

A  *     2    7 

^SSp    W>  sin2 P=P    "' 

log  sin2  7— log  sin2  7' 

0.4 
The  following  numerical  example  may  be  given:  — 

Let  I  =  22°,  sin  I  =  .37, 

If  =  18°,  sin  If  =  .31, 

log  .1369-log  .0961 
A  m  =  — =  .38. 

As  stated  before,  there  are  four  positions  in  which  equality 
of  light  is  produced,  each  of  which  must  be  read.  Therefore  the 
circle  is  graduated  into  four  quadrants  in  such  a  way  that  the 
angles  can  be  read  from  it  directly.  The  image  has  its  maximum 
brightness  when  the  two  Nicols  are  parallel.  This  position  is 
then  marked  90°  on  the  circle,  since  sin2  90°  =  1.  The  other 
positions  for  equality  will  be  180°  -  /,  180°  +  7,  360°  -  7. 

This  photometer,  as  we  have  said,  has  been  extensively  used 
abroad.  The  entire  work  of  the  Potsdam  Photometric  Durch- 
musterung  of  the  Northern  Heavens  was  carried  on  by  Miiller 
and  Kempf  with  instruments  of  this  design.  We  may  therefore 
appropriately  give  here  some  account  of  this  important  work. 
Its  purpose  is  to  furnish  the  photometric  magnitudes  of  all  of 
the  stars  in  the  BD.  down  to  magnitude  7.5.  Each  volume  con- 
tains the  stars  of  a  single  zone;  the  first  zone  extending  from 
0°  to  20°  declination,  the  second  from  20°  to  40°,  the  third  from 
40°  to  60°,  and  the  fourth  from  60°  to  90°.  A  fifth  volume  con- 
tains the  general  catalogue  of  14,199  stars.  It  was  begun  in 
1886  and  the  introduction  to  the  first  volume,  published  in 
1894,  contains  an  account  of  the  projected  plan.  Since  the 
brightnesses  of  the  stars  are  only  relative,  and  the  unit  to  which 
they  are  referred  is  entirely  arbitrary,  it  seemed  desirable  to 
follow  the  example  of  the  English  and  American  observers  and 
adopt  Polaris  as  the  standard.1  This  course  however  was  open 
to  the  objection  that  Polaris  would  frequently  differ  very  widely 
1  Potsdam  Phot.  DM.,  I,  7. 


VISUAL  PHOTOMETRY  121 

in  altitude  from  the  star  under  observation,  and  hence  a  large 
correction  for  atmospheric  absorption  would  be  required;  it 
was  therefore  decided  to  adopt  a  large  number  of  fundamental 
stars  suitably  distributed,  compare  them  exhaustively  with  the 
pole  star  and  among  themselves,  and  then  use  them  in  making 
the  actual  observations  of  other  stars.  They  were  selected  so 
as  to  lie  in  three  zones,  having  declinations  10°,  30°,  and  60°, 
at  intervals  of  30  m.  in  right  ascension,  to  range  in  magnitude 
from  5.0  to  6.7,  and  to  include  144  stars,1  which  were  combined 
in  432  pau*s  in  making  the  inter-comparisons.  Corrections  were 
applied  for  the  atmospheric  absorption.  The  colors  were  also 
compared,  and  numbered  according  to  the  following  scale; 
white  =  1,  yellowish  white  =  2,  whitish  yellow  =  3,  yellow  =  4, 
reddish  yellow  =  5,  yellowish  red  =  6,  red  =  7.2  Careful  inves- 
tigations were  made  to  discover  if  there  were  any  systematic 
errors  between  the  results  of  the  two  observers,  Miiller  and 
Kempf,  depending  either  upon  the  extent  of  the  difference  in 
magnitude  between  any  two  stars  of  a  pan*,  or  arising  from  a 
difference  in  color.  The  results  showed  that  while  a  relation 
was  apparent  the  errors  were  too  small  to  injure  the  results. 
Hence  all  of  the  later  determinations  depended  upon  the  mean 

i  (M  -  *).» 

The  photometer,  it  should  be  remembered,  gives  only  differ- 
ences in  magnitude  between  pairs  of  stars,  and  it  remained  to 
deduce  the  final  magnitudes  from  the  observed  differences. 
Since  there  were  432  combinations,  this  had  to  be  done  by  a 
series  of  approximations,  assuming  arbitrary  initial  values  for 
the  magnitudes  of  the  individual  stars.  The  values  adopted 
were  taken  from  the  Durchmusterung  of  Argelander,  first  be- 
cause they  were  entirely  independent  of  determinations  at 
Potsdam,  and  secondly  because  the  432  equations  give  the 
differences  in  brightness  only,  and  the  absolute  system  can  be 
determined  arbitrarily.  Furthermore,  the  BD.  is  much  the 
most  complete  catalogue  of  star  magnitudes  and  will  be  for 
a  long  time  an  almost  indispensable  source  for  studies  in 

1  Potsdam  Phot.  DM.,  i,  17.          2  Loc.  tit.,  109.          •  LOG.  rit..  111. 


122         THE  STUDY  OF  VARIABLE  STARS 

stellar  brightness;  hence  it  is  desirable  that  new  photometric 
catalogues  should  so  far  as  possible  be  connected  with  its 
system.  An  inspection  of  the  table  which  gives  the  differences 
Potsdam  —  ED.  shows  that  their  greatest  values  are  —  0.89 
and  +0.76,1  but  these  large  deviations  may  well  be  due  to  errors 
in  the  DM.  itself.  The  mean  difference  is  0.27.  The  mean  of 
the  star  magnitudes  for  both  systems  is  6.02,  so  that  at  mag- 
nitude 6.0  the  two  agree  exactly. 

The  exactness  of  the  Potsdam  magnitudes  is  indicated  by 
the  fact  that  the  relative  brightnesses  of  the  fundamental  stars 
are  correct  to  0.05  mg.  A  correction  to  the  whole  system  may 
be  made  at  any  time  if  it  should  seem  desirable  to  change  the 
standard.  Therefore  the  material  afforded  by  the  magnitudes 
of  the  standard  stars  is  very  exact  and  homogeneous. 

The  next  photometer  to  be  considered  will  be  that  designed 
by  Pickering  and  known  as  the  meridian  photometer.  It  is 
described  at  length  in  Annals,  H.C.O.,  vol.  14,  and  later  modi- 
fications are  given  in  volume  23.  In  this  form  of  photometer  the 
principle  of  polarized  light  is  used,  but  in  quite  a  different  form 
from  that  utilized  in  the  Zollner  instrument,  as  the  following 
preliminary  statement  will  show. 

In  place  of  using  a  Nicol  prism  as  a  polarizer,  which  permits 
only  one  beam  of  light  to  pass  through  it,  another  form  of 
double  image  prism  is  used  which  permits  two  beams  of  light, 
polarized  in  planes  at  right  angles  to  each  other,  to  pass  through 
the  analyzer;  as  the  prism  is  rotated,  one  image  diminishes 
while  the  other  increases,  and  when  the  point  of  equality  is 
reached,  the  circle  is  read.  Instead  of  an  artificial  star  the  pole 
star  or  some  other  close  circumpolar  is  used  for  comparison,  and 
the  difference  in  magnitude  can  be  determined  directly  from  the 
circle  readings.  A  description  of  the  instrument  is  given  below. 
An  excellent  drawing2  of  it  can  be  found  in  Young's  General 
Astronomy,  an  adaptation  from  which  is  given  here. 

Two  lenses,  10.5  cm.  in  diameter,  and  of  the  same  focal  length, 

1  Potsdam  Phot.  DM.,  I,  116,  Table  v. 

*  C.  A.  Young,  Gen.  Ast.,  ed.  1891,  473.  Ginn  &  Company,  publishers. 


VISUAL  PHOTOMETRY 


123 


are  placed  side  by  side  in  the  same  tube,  which  is  set  horizontal 
in  an  east  and  west  line.  In  front  of  each  is  set  a  plane  mirror 
at  an  angle  of  45°,  adjusted  in  such  a  way  that  it  reflects  into 
the  tube  stars  which  are  on  or  very  near  the  meridian.  The 
mirror  of  one  is  adjusted  so  as  always  to  send  the  light  of  the 
pole  star  through  one  lens,  and  the  other  can  be  rotated  about 
an  axis  in  such  a  way  as  to  reflect  stars  on  any  part  of  the 
meridian  into  the  second  lens.  The  latter  can  also  be  turned  by 
rods  and  made  to  reach  a  star  of  any  declination  a  quarter  of 
an  hour  before  or  after  its  meridian  passage.  The  mirror  which 
serves  to  reflect  the  pole  star  is  also  capable  of  adjustment  and 
can  bring  it  into  any  part  of  the  field.  The  two  lenses  are  some- 


/ 


Figure  19 


PICKERING'S  MERIDIAN  PHOTOMETER 


what  inclined  so  that  their  optical  axes  are  not  parallel,  but  lie 
at  such  an  angle  that  their  images  would  be  formed  at  the  same 
point.  Just  before  this  point  is  reached  a  double  image  prism, 
D,  made  of  Iceland  spar  compensated  by  glass,  the  object  of 
which  is  to  bring  the  two  pencils  of  light  together,  is  inserted 
in  the  path  of  the  rays.  Each  of  the  two  beams  of  light  falling 
upon  it  is  separated  into  two,  which  are  polarized  at  right 
angles  to  each  other.  The  two  outer  beams,  ae  and  60,  i.e.,  a 
extr.  and  6  ord.,  are  cut  off  by  a  diaphragm,  and  the  two  inner 
beams  are  made  to  coincide,  and  pass  through  a  Nicol  prism, 
which  may  be  rotated  and  the  amount  of  rotation  read  from  a 
scale,  C.  They  then  enter  the  eyepiece,  where  the  observer 
sees  two  star  images,  one  coming  from  Polaris,  and  one  from 
the  star  to  be  compared  with  it.  The  images  will  consist  of 


124          THE  STUDY  OF  VARIABLE  STARS 

light  polarized  in  planes  at  right  angles  to  each  other,  and  con- 
sequently, when  the  Nicol  is  rotated,  one  image  will  diminish 
in  brightness,  while  the  other  increases,  and  there  will  be  four 
positions  in  which  they  will  be  equal.  Much  care  is  required 
in  adjusting  the  different  parts  of  the  apparatus  in  order  to 
obtain  the  desired  results,  the  details  of  which  are  fully  de- 
scribed in  Annals,  14,  1-8.  The  process  of  observing  is  as  fol- 
lows. Two  persons  are  required  in  the  work,  the  observer,  "A," 
and  the  recorder,  "B."  A  dark  curtain  cuts  off  the  light  of  the 
room  from  the  eyepiece  where  the  observer  sits,  while  the 
recorder  is  placed  at  the  side  of  the  table,  with  a  sidereal  clock 
and  all  necessary  papers  before  him.  The  observer  "A"  brings 
the  pole  star  into  the  middle  of  the  field  by  moving  the  northern 
prism  with  adjusting  rods.  The  recorder  "B"  selects  the  star 
to  be  compared  in  accordance  with  a  plan  previously  made, 
and  turns  the  southern  prism  so  as  to  bring  this  star  also  into 
the  field.  "A"  then  places  the  image  of  Polaris  near  that  of 
the  other  star,  and  turns  the  Nicol  until  the  two  images  are 
equally  bright.  The  recorder  then  reads  the  circle  to  tenths  of 
a  degree,  the  observer  turns  the  Nicol  still  farther  until  the 
images  are  again  equal,  and  the  recorder  takes  the  reading. 
The  observer  then  turns  the  northern  prism,  which  throws  in 
the  image  of  Polaris  so  that  it  appears  on  the  other  side  of  the 
second  star,  and  makes  two  more  settings.  This  last  change  is 
made  in  order  to  eliminate  a  personal  error  arising  from  a  right 
and  left  comparison,  the  possibility  of  which  was  earlier  dis- 
covered by  Pickering  when  making  similar  observations  of 
lapetus  (Annals,  9,  222).  The  time  consumed  in  making  each 
observation  is  one  minute.  As  the  observations  are  pretty 
continuous,  one  star  being  brought  into  the  field  as  soon  as 
another  has  been  observed,  a  series  is  limited  to  about  an  hour. 
The  method  of  reduction  will  be  slightly  different  from  that 
adapted  to  the  Nicol  prism  and  the  Zollner  photometer,  for 
here  we  have  the  case  of  two  images  in  which  the  light  is  polar- 
ized in  planes  at  right  angles  to  each  other.  Hence  the  intensity 
of  one  will  be  A  cos2  /  and  B  sin2  7,  where  A  and  B  are  the  in- 


VISUAL   PHOTOMETRY  125 

tensities  of  the  incident  rays,  i.e.,  the  brightnesses  of  the  two 
stars.  If  I  is  the  angle  counted  from  the  position  where  the 
image  of  Polaris  disappears,  then  A  is  its  brightness,  and  B  is 
the  brightness  of  the  other  star.  Then,  since  the  images  are 
equal,  we  shall  have 

A  cos2  /  =  B  sin2  /, 

and  —  =  tan2  7, 

whence,  by  Pogson's  rule, 

A=     Aro 

j  log  tan2  7 

and  A  m  =  — 2— . 

0.4 

Further  details  of  the  computation  and  the  construction  of 
tables  for  facilitating  it  may  be  found  described  in  the  Annals. 
The  second  prism  is  turned  so  as  to  bring  the  image  of  Polaris 
into  the  field  at  the  beginning,  middle,  and  end  of  a  series,  in 
order  to  see  if  the  two  prisms  are  in  good  adjustment.  There 
are  three  errors  to  which  the  star  magnitudes  are  subject;  the 
first  may  result  from  a  possible  difference  in  the  images  formed 
by  the  two  object  glasses,  the  second  from  atmospheric  absorp- 
tion, and  the  third  from  any  possible  deviation  of  the  magnitude 
of  Polaris,  which  is  assumed  to  be  2.0.  All  of  these  sources  of 
error  were  investigated,  and  material  collected  and  prepared 
for  making  corrections  for  them.  In  view  of  the  recent  discov- 
ery of  the  variation  of  Polaris  it  is  interesting  to  inquire  if  it 
was  revealed  by  this  investigation.  Pickering  states  that  since 
all  of  the  observations  were  reduced  by  means  of  the  pole  star, 
the  residuals  of  the  standard  stars  would  show  whether  it 
varied  or  not.  But  from  a  study  of  them  the  conclusion  was 
drawn  that  it  did  not  undergo  a  variation  of  long  period.  After 
the  variability  became  known  and  its  period  was  found  to  be 
3.9683  days,  the  residuals  were  grouped  according  to  the  phase 
and  the  means  taken.  The  evidence  of  variation  was  unmis- 
takable and  the  form  of  the  light  curve  was  easily  obtained. 


126          THE  STUDY  OF  VARIABLE  STARS 

With  a  smaller  instrument  similar  to  this  the  first  volume  of 
the  Harvard  Photometry  was  made,  and  published  in  1884.  Its 
purpose  was  the  observation  of  stars  not  fainter  than  the  sixth 
magnitude  lying  between  the  pole  and  30°  south  declination. 
In  preparing  the  list  the  various  catalogues  and  charts  con- 
structed to  include  such  stars  were  carefully  consulted,  namely 
Argelander's  Uranometria  Nova,  Heis's  Atlas  Coelestis,  Gould's 
Uranometria,  and  the  Durchmusterung,  from  which  all  stars 
not  fainter  than  6.5  mg.  were  taken.  The  final  catalogue  con- 
tains 4260  stars.  At  a  later  period  a  second  meridian  photom- 
eter was  constructed,  having  larger  lenses,  and  intended  for 
determining  the  magnitudes  of  fainter  stars.  For  this  reason, 
since  Polaris  was  too  bright  for  purposes  of  comparison,  X 
Ursae  Minoris  was  used  as  the  standard  star. 

The  object  was  to  determine  the  magnitudes  of  stars,  chiefly 
of  the  9th  magnitude  and  brighter,  20,125  in  number,  in  zones 
20'  wide  at  intervals  of  5°,  from  —  20°  to  the  north  pole.  Vari- 
ous other  volumes  containing  observations  made  with  these 
two  instruments  have  been  issued  by  the  Harvard  College 
Observatory.  Altogether  they  form  a  most  valuable  contribu- 
tion to  the  study  of  stellar  photometry. 

There  are  several  other  photometers  which  make  use  of 
polarization  apparatus,  but  the  two  described  are  the  best 
known  and  have  been  most  extensively  used.  A  third  pho- 
tometer, designed  by  Pickering,  makes  use  of  an  artificial  star, 
the  light  of  which  is  cut  down  by  a  photographic  wedge  in  order 
to  equal  that  of  the  real  star.  The  accompanying  figure  shows 
its  construction,  which  in  general  outline  is  similar  to  that  of 
the  Zollner  photometer. 

The  tube  which  is  attached  to  the  end  of  the  telescope  bears 
at  right  angles  to  it  an  arm  containing  the  artificial  star.  The 
light  from  an  incandescent  lamp,  L,  passes  through  a  small 
aperture  in  the  diaphragm,  Z),  then  through  a  condensing  lens, 
P,  which  projects  it  upon  a  plate  of  plane  parallel  glass,  B9 
from  both  surfaces  of  which  it  is  reflected,  forming  images  in 
the  focal  plane  of  the  objective  at  E  and  F,  on  either  side  of  the 


Figure  20 

WEDGE  PHOTOMETER,  YERKES  OBSERVATORY 


128          THE  STUDY  OF  VARIABLE  STARS 

true  star,  H.  The  reduction  of  light  is  effected  by  the  interposi- 
tion of  the  wedge,  W,  movable  by  the  rack  and  pinion  R,  the 
position  of  which  can  be  read  on  the  scale.  Various  devices  are 
introduced  for  the  purpose  of  making  the  artificial  star  resemble 
as  closely  as  possible  the  true  star  in  size  and  sharpness;  among 
them  a  ground  glass,  G,  close  to  which  is  a  piece  of  blue  glass, 
to  render  the  artificial  star  less  yellow,  and  a  pair  of  shade 
glasses  at  A.  The  diaphragm  D  is  adjustable,  and  has  more 
than  one  aperture.  The  photographic  wedge  consists  of  a  regu- 
lar photographic  film,  prepared  in  the  manner  described  in 
Chapter  VII  at  the  Harvard  Observatory.  A  thin  plate  of  glass 
is  fastened  to  it  for  protection.  In  making  the  observation  the 
telescope  is  moved  until  the  image  of  the  real  star  lies  between 
the  two  images  of  the  artificial  star.  The  wedge  is  then  moved 
until  equality  is  secured,  and  the  reading  of  the  scale  is  recorded. 
As  before,  comparison  must  be  made  with  a  standard  star, 
whose  magnitude  is  already  known.  The  scale  reading  for  its 
equality  with  the  artificial  star  is  found,  and  the  difference  in 
readings,  when  multiplied  by  the  value  of  one  division  on  the 
scale,  will  equal  the  difference  in  magnitudes.  It  remains,  then, 
to  investigate  the  value  of  one  division,  or  as  it  is  termed,  the 
wedge  constant.  This  problem  is  fully  treated  by  Parkhurst,1 
who  gives  the  results  of  his  researches  on  an  instrument  at  the 
Yerkes  Observatory,  in  the  course  of  which  he  employed  sev- 
eral different  methods.  The  one  most  easily  understood  is  the 
observation  of  a  series  of  standard  stars  whose  magnitudes  are 
well  known,  such  as  the  Pleiades.  The  wedge  should  also  be 
calibrated,  that  is,  an  investigation  should  be  made  to  find  if 
the  value  of  a  division  on  the  scale  is  constant  throughout  the 
entire  length.  This  form  of  instrument,  attached  to  a  fifteen- 
inch  equatorial,  has  been  used  extensively  at  the  Harvard 
Observatory,  particularly  for  the  observations  of  long  period 
variables,  both  in  finding  the  magnitudes  of  the  fainter  com- 
parison stars  and  in  making  observations  of  the  variable.  It 
has  been  used  by  Parkhurst  at  the  Yerkes  Observatory,  with 
1  Ap.  J.,  13,  249. 


VISUAL  PHOTOMETRY  129 

both  the  six-inch  and  the  forty-inch  telescopes.  It  has  also 
been  employed  elsewhere  in  this  country,  and  has  been  found 
to  give  satisfactory  results. 

In  the  foregoing  pages  no  attempt  has  been  made  to  discuss 
fully  any  of  the  instruments  under  consideration  or  the  errors 
to  which  each  is  subject.  The  purpose  has  been  rather  to  give 
the  reader  the  idea  of  their  general  construction  and  use,  leav- 
ing the  original  sources  to  be  taken  up  by  the  student  who  is 
specializing  in  such  work.  A  fairly  full  treatment  of  polarized 
light  has  been  given,  since  an  understanding  of  its  principles 
is  necessary,  and  no  textbook  in  Astronomy  adequately  dis- 
cusses it. 

No  mention  has  been  made  of  the  method  of  extinction  by 
which  photometric  observations  can  be  made.  There  are  sev- 
eral photometers  of  this  kind,  in  which  the  star's  light  is 
diminished  until  it  disappears,  the  point  of  extinction  being 
read  on  the  scale.  In  this  case,  as  before,  recourse  must  be  had 
to  a  standard  star,  which  is  observed  in  the  same  way.  This 
may  be  accomplished  by  cutting  down  the  aperture  of  the 
object-glass,  or  by  inserting  at  the  eye  end  of  the  telescope  a 
wedge  or  some  kind  of  polarizing  apparatus.  This  method  is 
not  now  in  general  favor,  as  the  observation  is  supposed  to  be 
more  fatiguing  to  the  eye  of  the  observer,  and  the  instant  of 
extinction  is  not  clearly  denned.  However,  some  astronomers 
find  no  difficulty  in  making  the  observation,  and  such  instru- 
ments are  still  in  use  and  giving  good  results.  The  writer,  on  a 
recent  visit  to  the  Capodimonte  Observatory,  in  Naples,  found 
that  the  Director,  Professor  Bemporad,  was  using  a  Toepfer 
extinction  photometer  with  much  success.  On  being  asked  if 
the  observation  was  not  difficult  and  trying  to  the  eyes  he 
replied  decidedly  in  the  negative.  The  results  of  his  work  have 
not  yet  been  published. 


CHAPTER  VII 

PHOTOGRAPHIC  PHOTOMETRY 

THE  subject  of  star  color  and  its  effect  upon  visual  photo- 
metric measures  has  been  referred  to  several  times  in  the  pre- 
ceding chapters,  but  it  becomes  still  more  important  when  the 
study  of  photographic  photometry  is  taken  up.  Hence  a  dis- 
cussion of  it  should  precede  the  consideration  of  the  general 
subject  of  this  chapter.  Mention  has  been  made  of  Chandler's 
scale  in  connection  with  the  colors  of  variable  stars,  and  of  the 
Potsdam  scale  in  the  description  of  the  Potsdam  Photometric 
Durchmusterung.  These  two  are  not  the  only  scales.  Argelander 
first  suggested  a  chromatic  scale,  which  consisted  of  four  colors, 
red,  orange,  yellow,  white;  but  he  realized  that  his  own  eye  was 
not  very  sensitive  to  color  impressions,  and  made  very  little 
systematic  study  of  star  colors. 

In  1872  Schmidt,1  from  the  observatory  at  Athens,  wrote  to 
the  Nachrichten  an  account  of  the  results  of  his  experiments  in 
the  study  of  color  with  different  telescopes  in  different  localities. 
He  came  to  the  conclusion  that  within  certain  limits  color  can 
be  expressed  by  a  numerical  scale.  From  his  series  he  excludes 
such  colors  as  green,  blue,  and  purple,  which  are  seen  in  the 
components  of  double  stars,  and  also  the  greenish  shimmer 
which  many  isolated  stars  have,  and  limits  himself  to  the 
orange,  or  colors  which,  beginning  with  pure  white,  pass 
through  all  the  stages  of  yellow  and  finally  emerge  into  red. 

In  my  experience  neither  a  pure  white  nor  a  decidedly  red  star 
occurs.  In  all  the  so-called  white  stars,  such  as  Sirius  and  Vega,  I  find 
some,  though  very  little,  mixture  of  yellow.  In  all  the  red  stars,  with- 
out exception,  the  fundamental  color  is  an  intensive  yellow,  with  a 
decided,  though  unequally  strong,  inclination  toward  red.  This  is  the 
case  with  Antares.  A  true  red,  carmine,  or  blood  red,  a  red  such  as  I 
know  in  the  protuberances,  the  red  of  the  Fraunhofer  line  C,  I  have 

*  A.N.  1897. 


PHOTOGRAPHIC  PHOTOMETRY  131 

never  found  in  the  case  of  a  star.  Therefore  in  my  scale  I  make  pure 
white  0,  and  the  true  red,  without  any  mixture  of  yellow,  I  value  10. 
Between  these  two  lies  the  bright  yellow  at  4,  the  intense  golden  yel- 
low at  6,  and  all  my  red  stars  have  numbers  between  6.5  and  9. 

The  rest  of  his  paper  contains  interesting  remarks  on  the 
color  of  variable  stars,  which  belong  more  properly  to  the 
chapter  on  the  statistical  study. 

Other  observers  adopted  scales  of  the  same  sort,  but  assigned 
different  numbers  to  the  different  shades  of  color.  They  found 
that  the  estimations  of  the  color  were  affected  by  moonlight, 
twilight,  dust,  or  cloud,  though  Schmidt  thought  only  the 
twinkling  on  the  horizon  need  be  regarded.1  The  instrument 
also  has  an  effect  on  the  color.  One  observer  found  that  com- 
parisons made  with  a  reflector  and  a  refractor  differed,  as  the 
latter  has  chromatic  aberration,  while  the  former  has  not. 
Argelander  says  that  a  certain  star  will  appear  brighter  in  con- 
trast to  a  white  star  the  larger  the  light  gathering  power  of  the 
telescope  which  is  used  to  observe  them.  This  is  explained 
by  the  Purkinje  phenomenon,  which  was  described  in  the  pre- 
ceding chapter.  Most  of  the  earlier  comparisons  were  merely 
eye  estimates.  Chandler 2  makes  some  interesting  general  re- 
marks in  connection  with  a  paper  on  the  colors  of  variables,  in 
the  introduction  of  which  he  says :  — 

I  had  long  been  impressed  with  the  importance  of  an  investigation 
of  the  sort  in  question,  but  had  been  deterred  from  undertaking  it  by 
the  difficulties,  physical  and  physiological,  in  devising  a  rational  and 
practical  method,  and  the  establishment  of  a  correct  color-scale. 

He  adopted  a  scale  similar  to  Schmidt's  though  not  coincid- 
ing with  it,  in  which  0  corresponded  to  white,  2  to  yellow,  4  to 
full  orange,  and  hence  up  to  10,  which  is  full  red,  such  as  we  find 
in  stars  like  S  Cephei  and  R  Leporis.  His  remarks  upon  its 
adoption  are  interesting  and  illuminating. 

It  is  freely  admitted  that  there  is  much  vagueness  in  this  descrip- 
tion, as  well  as  in  the  mental  picture  of  the  imaginary  standards  to 
which  it  was  sought  to  refer  the  estimates.  The  difficulty  is  inherent 

1  J.  G.  Hagen,  S.J.,  Ver.  St.,  178.  2  Ast.  Jour.,  8, 137. 


132          THE  STUDY  OF  VARIABLE  STARS 

and  has  been  experienced  by  other  observers.  Indeed,  in  the  begin- 
ning, before  confidence  had  been  acquired  by  practice,  I  strongly 
doubted  whether  the  method  would  yield  results  to  be  depended  upon; 
but  on  further  acquaintance  I  am  convinced  that  the  certainty  of  the 
process  of  mental  reference  of  color-impressions  to  imaginary  stand- 
ards, and  the  fixedness  of  the  latter,  are  greater  than  would  be  natu- 
rally inferred  by  an  observer  previous  to  trial. 

In  the  next  part  of  his  article  Chandler  discusses  a  plan  by 
which  he  made  use  of  Argelander's  step  method  in  converting 
difference  in  color  between  two  stars  into  difference  in  bright- 
ness by  interposing  a  shade  of  colored  glass,  which  by  selective 
absorption  altered  the  apparent  relative  brightness  of  stars  of 
different  colors.  Thus  a  red  star  which  appears  exactly  equal 
to  a  white  star  when  viewed  in  the  ordinary  way,  appears 
fainter  than  the  latter  when  a  blue  shade  glass  is  applied  to  the 
eyepiece,  and  brighter  when  a  red  one  is  used. 

These  differences,  which  can  be  estimated  very  precisely  by  Arge- 
lander's  method,  thus  become  measures  of  the  difference  of  color,  of 
course  on  an  entirely  arbitrary  scale,  depending  on  the  amount  and 
character  of  the  selective  absorption  of  the  shades  employed. 

He  makes  use  of  his  scale  for  determining  the  redness  of  the 
variables,  and  comes  to  practically  the  same  conclusion  which 
Schmidt  reached,  viz.,  that  the  color  varies  with  the  length 
of  period. 

Osthoff,1  at  Cologne,  worked  twenty-five  years  on  star  colors, 
and  in  1900  published  the  results  of  his  investigation,  including 
a  catalogue  of  the  colors  of  1009  stars.  The  description  of  his 
method  can  best  be  given  in  his  own  words:  — 

The  observing  room  was  always  entirely  darkened.  I  covered  my 
head  and  the  eye  end  of  the  telescope  with  a  dark  cloth.  The  observ- 
ations were  written  down  in  the  dark,  and  the  color  was  always  ex- 
pressed in  one  figure.  Only  under  the  most  pressing  circumstances 
was  the  lantern  opened  during  the  time  of  observation,  and  then  only 
to  look  at  the  star  chart.  I  always  estimated  an  unknown  star  with 
reference  to  a  known  star.  At  the  conclusion  of  the  observations,  still 
during  the  night,  or  at  the  latest  the  next  morning,  I  glanced  over  my 

1  A.N.  3657-58. 


PHOTOGRAPHIC  PHOTOMETRY  133 

notes  and  identified  the  stars.  Before  the  beginning  of  each  evening's 
work  I  looked  over  the  program,  but  did  not  take  any  heed  of  the  ob- 
servations already  made. 

When  there  was  bright  moonlight,  unsteady  air,  or  a  too  cloudy 
sky,  I  made  no  estimation  of  the  color.  I  looked  long  and  fixedly  at 
each  star  until  the  impression  of  its  color  no  longer  fluctuated.  I 
obtained  for  the  mean  duration  of  a  color  estimate  2.21  minutes. 

He  adopted  Schmidt's  scale,  but  separated  the  classes  more 
definitely,  and  extended  the  scale  on  either  end  to  —  lc  brilliant 
pure  white,  and  12C  dark  pure  red.  He  then  adds  the  following 
remarks,  which  are  most  valuable  in  view  of  his  long  experience 
in  this  kind  of  work.  They  refer  to  its  subjective  side :  — 

Having  applied  this  scale  to  many  stars,  through  thousands  of  es- 
timations I  have  tested  its  practicability  and  found  that  it  expresses 
accurately  the  relations  among  the  stars  as  long  as  they  are  sufficiently 
bright.  If  their  light  is  too  faint,  white  changes  into  gray,  etc.,  yel- 
low into  brown,  and  orange  into  reddish  brown.  The  mass  of  the 
fainter  stars  shine  in  a  monotonous  gray.  The  colors  blue  and  green 
are  subjective.  They  arise  when  the  observer  has  not  protected  him- 
self from  the  influence  of  outside  light,  such  as  the  shimmer  of  the 
starlit  heavens,  reflections  from  the  metallic  parts  of  the  instrument, 
or  moonlight.  These  remarks  are  not  to  be  applied  to  the  vivid  blue 
color  of  the  smaller  component  of  a  double  star. 

Enough  has  been  said  to  show  that  such  elusive  impressions 
as  star  colors  can  be  classified  accurately  when  sufficient  experi- 
ence has  been  acquired.  Several  other  determinations  of  star 
color  have  been  made,  and  catalogues  published.  In  the  latter 
part  of  his  paper  Osthoff  compares  the  scales  of  the  different 
observers,  and  arrives  at  an  expression  of  their  differences.  He 
also  studied  his  own  results  from  several  different  points  of 
view,  but  it  is  not  necessary  for  our  purpose  to  carry  the  matter 
farther  in  this  direction,  for  recent  investigations  have  so  per- 
sistently pointed  to  the  connection  between  color  and  spectral 
type  that  the  interest  is  centering  now  on  the  latter  factor 
rather  than  the  former.  However,  before  passing  to  a  consid- 
eration of  it,  it  is  interesting  to  note  that  while  different  color- 
imeters, or  instruments  for  determining  the  color,  such  as  the 
quartz  section  in  the  Zollner  photometer,  have  been  devised 


134          THE  STUDY  OF  VARIABLE  STARS 

and  successfully  applied,  no  catalogues  of  star  colors  based 
upon  their  use  have  ever  been  published.  As  Hagen1  says, 
"  Colorimetry  has  lagged  far  behind  photometry." 

That  there  is  a  relation  between  the  color  of  a  star  and  its 
spectrum  is  evident  theoretically  from  a  consideration  of  the 
classification  of  stellar  spectra;  and  it  is  also  obvious  when  we 
consider  how  the  spectrum  is  formed.  The  light  of  the  star 
falling  upon  a  prism  is  broken  up  into  its  component  colors. 
The  lines  in  its  spectrum  cut  out  certain  portions  of  these 
colors,  and  there  is  frequently  general  absorption.  Hence  we 
can  easily  see  that  the  colors  which  are  left,  when  combined, 
will  not  give  white  light,  but  the  resulting  tone  will  depend 
upon  the  mixture.  Thus  we  should  hardly  expect,  as  Schmidt 
evidently  did,  to  find  a  star  which  had  color  like  the  pure  red 
of  the  C  line.  Such  a  star  might  occur  if  that  line  alone  were 
present  or  if  it  were  the  dominant  color  in  the  spectrum. 
Neither  should  we  look  for  blue  stars,  for  their  existence  would 
require  absorption  in  the  red  end  of  the  spectrum.  But  absorp- 
tion comes  in  from  the  other  end;  hence  the  blue  end  is  cut  off 
first.  The  only  stars  likely  to  have  the  blue  color  predominat- 
ing in  the  spectrum  are  those  of  the  fifth  type,  which  have 
bright  bands  at  X  4688  and  4633.  Their  color  can  be  gauged 
by  remembering  that  they  lie  about  half  way  between  the  F  line 
and  the  G  group  in  the  solar  spectrum,  and  hence  will  be  a 
bright,  rather  light  blue,  but  a  decided  blue,  with  no  tinge  of 
another  color.  Unfortunately  these  stars  are  rather  faint,  and 
none  of  them  are  found  in  the  catalogues  of  colored  stars.  The 
best  list  of  stars  of  the  fifth  type  was  prepared  by  Mrs.  Fleming, 
and  is  found  in  Annals,  H.C.O.,  vol.  56,  no.  vi.  Only  four  stars 
on  this  list  in  the  northern  heavens  are  bright  enough  to  be 
found  in  the  PD.,  and  their  colors  as  given  there  are  W+  or 
GW.  Barnard  gives  some  interesting  facts  in  regard  to  the 
focus  and  color  of  certain  temporary  stars.  In  writing  of  Nova 
Lacertae 2  in  1911,  he  states  that  it  had  two  distinct  foci.  At 
one  of  them  the  image  had  but  little  color  and  was  surrounded 
1  Ver.  St.,  291.  *  A.N.  4468. 


PHOTOGRAPHIC  PHOTOMETRY  135 

by  a  reddish  glow.  The  other  image  was  of  a  beautiful  crimson 
color,  surrounded  with  a  greenish-gray  glow,  and  was  in  focus 
8  mm.  farther  from  the  object  glass.  He  explains  it  as  being 
due  to  the  great  brilliance  of  the  Ha  line.  This  star  would  seem, 
then,  to  be  an  illustration  of  Schmidt's  long  sought  for  type  10, 
of  which  he  says:  "A  true  red,  carmine  or  blood  red,  a  red  such 
as  I  know  in  the  protuberances,  the  red  of  the  Fraunhofer  line 
C,  I  have  never  found  in  the  case  of  a  star." 

It  is  a  curious  and  interesting  fact  that  both  Schmidt  and 
Osthoff  rather  avoided  the  subject  of  the  color  of  double  stars, 
but  seemed  to  take  it  for  granted  that  among  these  at  least, 
stars  of  a  true  blue  were  to  be  found.  While  the  discussion  of 
this  point  does  not  bear  especially  on  the  subject  before  us,  it 
may  be  admitted  perhaps,  on  account  of  its  very  great  interest. 
An  investigation  of  the  subject  was  recently  made  by  Mr.  Louis 
Bell,1  who  approached  it  from  the  point  of  view  of  physiological 
optics.  He  was  led  to  do  so  because  of  the  great  variety  and 
bizarre  array  of  colors  assigned  to  the  components  of  double 
stars,  such  as  may  be  found  in  any  of  the  English  books  con- 
taining lists  of  these  objects.  Webb,  for  example,  in  his  Celestial 
Objects  for  Common  Telescopes,  uses  such  adjectives  as  lilac, 
mauve,  cool  gray-green,  ashy  yellow,  smalt  blue,  topaz,  fawn 
color,  etc.,  colors  never  found  in  isolated  stars.  An  examination 
of  the  use  of  these  names  shows  that  they  are  applied  in  nearly 
every  case  to  the  smaller  component  of  a  pair,  which  may  be 
indiscriminately  an  optical  or  a  physical  double.  This  fact  was 
long  considered  to  indicate  that  the  smaller  star  was  not  so  far 
advanced  in  evolution  as  the  brighter  component,  and  hence 
would  show  a  spectrum  of  an  earlier  type;  but  even  such  a 
spectrum  would  not  give  a  blue  color,  since  stars  of  the  early 
type  are  pure  white,  or  a  pale  yellowish  white.  Besides,  this 
certainly  could  not  be  regarded  as  a  valid  explanation  when 
we  consider  the  fact  that  such  colored  doubles  are  not  always 
binary  systems,  but  may  be  merely  optical  doubles,  in  which, 
while  the  two  stars  are  in  the  same  line  of  sight,  one  is  far  dis- 
i  Ap.  J.t  31, 234. 


136          THE  STUDY  OF  VARIABLE  STARS 

tant  from  the  other,  so  that  there  can  be  no  possibility  of  a 
physical  connection  between  them. 

It  has  long  been  known  that  contrasts  in  color  are  in  some 
way  related  to  the  difference  in  magnitude.  Struve  found  an 
average  difference  of  about  0.5  mg.  for  doubles  of  exactly  the 
same  color,  a  difference  a  little  greater  than  a  magnitude  for 
doubles  which  had  slightly  different  colors,  but  a  difference  of 
over  two  magnitudes  when  the  colors  were  decidedly  unlike. 
The  study  of  the  spectra  of  a  few  doubles,  which  Bell  found  on 
the  Harvard  records,  where  the  spectra  of  both  components 
had  been  photographed,  showed  that  where  they  were  of  the 
same  spectral  type  they  did  not  differ  greatly  in  magnitude  or 
in  color.  Where  the  companion  was  of  an  earlier  type  than  the 
principal  star,  there  was  a  decided  contrast  in  color,  and  a 
greater  difference  in  magnitude;  such  as 

e  Bootis;  Star  A,  mg.  2.7,  K,  very  yellow; 

Star  B,  mg.  5.1,  A,  very  blue. 

The  letter  refers  to  the  spectral  type.  There  are  a  few  instances 
in  which  the  primary  has  the  earlier  spectral  type,  in  which 
case  there  is  again  a  contrast  in  the  colors;  e.g., 
ft'  Bootis;  Star  A,  mg.  4.5,  F,  flushed  white  or  yellow; 

Star  B,  mg.  6.5,  K,  greenish  white,  yellowish  azure. 
The  epithet  "yellowish  azure"  applied  to  a  star  of  type  K  at 
once  shows  that  the  subjective  element  in  estimating  the  color 
is  very  strong. 

The  possibility  of  this  was  not  ignored  by  the  earlier  workers 
in  double  stars,  but  it  was  dismissed  from  general  consideration 
for  two  reasons:  firstly,  the  colors  must  be  real,  because  they 
persisted  even  when  the  primary  was  hidden  by  an  occulting 
bar;  secondly,  if  such  colors  are  due  to  contrast  they  must  be 
complementary.  Bell's  paper  then  continues  with  an  exposi- 
tion of  the  various  physiological  causes  which  can  produce  such 
phenomena,  and  ends  with  an  account  of  experiments  made 
with  artificial  doubles  to  prove  them.  These  causes  are  familiar 
to  readers  who  are  proficient  in  the  subject  of  physiological 
optics,  and  can  only  be  mentioned  here.  They  are  "simultane- 


PHOTOGRAPHIC  PHOTOMETRY  137 

cms  contrast,"  "fatigue  color,"  the  "Purkinje  phenomenon," 
and  "dazzle  tints." 

The  preceding  pages  show  the  importance  of  color  in  making 
visual  observations  of  stellar  brightness.  We  shall  now  see  that 
it  has  an  equally  important  effect  upon  the  photographic  image 
of  a  star;  but  before  considering  this  point  it  will  be  necessary 
first  to  give  some  account  of  celestial  photography  in  general, 
including  some  practical  matters  regarding  telescopes. 

The  first  astronomer  to  suggest  that  the  size  of  the  photo- 
graphic image  of  a  star  would  vary  with  its  brightness  and  the 
duration  of  the  exposure,  was  Professor  G.  P.  Bond1  of  the 
Harvard  College  Observatory.  In  1858  he  published  an  article 
in  the  Astronomische  Nachrichten  on  Stellar  Photography,  in 
which  he  made  the  following  introductory  statement.  The 
entire  passage  is  quoted  because  it  shows  that  he  saw  the 
problem  clearly,  but  also  that  certain  technical  difficulties 
escaped  him. 

Photographs  of  Stars  of  unequal  brightness  present  marked  pecu- 
liarities in  size  and  intensity,  when  their  images  formed  in  equal  ex- 
posures are  compared  together,  at  once  suggesting  the  possibility  of 
classifying  them  according  to  a  scale  of  photographic  or  chemical 
magnitudes,  analogous  to  the  common  optical  scale,  but  differing  from 
it  essentially  in  the  fact  of  its  being  based  upon  actual  measurements, 
in  place  of  the  vague  and  uncertain  estimates  to  which  astronomers 
have  hitherto  resorted  in  attempting  to  express  with  numbers  the 
relative  brightness  of  different  stars. 

There  are  three  particulars  in  which  the  proposed  system  will  have 
an  unquestionable  advantage  over  that  in  common  use,  provided  that 
the  chemical  action  of  the  starlight  is  found  to  be  energetic  enough  to 
furnish  accurate  determinations  of  its  amount.  It  will  be  less  liable 
to  be  affected  by  individual  peculiarities  of  vision.  There  will  be  less 
room  for  discordance  between  different  observers,  or  for  disagreement 
between  the  conclusions  of  tha  same  observer  at  different  times,  as  to 
the  qualities  or  proportions  constituting  the  various  grades  of  magni- 
tude. —  Lastly  it  will  meet  perfectly  the  greatest  of  the  many  difficul- 
ties of  the  problem  —  the  comparison  of  stars  exhibiting  diversity  of 
color. 

Though  Bond  erred  in  regard  to  the  difficulty  presented  in 
i  A.N.  1158-59. 


138          THE  STUDY  OF  VARIABLE  STARS 

the  last  statement,  its  existence  can  hardly  be  considered  a  dis- 
advantage, because  while  diversity  of  color  has  added  to  the 
complexity  of  the  problem,  the  efforts  to  overcome  it  have 
greatly  added  to  our  knowledge. 

Bond  concluded  his  article  by  saying:  — 

There  seems  to  remain  in  the  way  of  obtaining  a  very  high  degree 
of  precision  by  those  means,  only  the  difficulty  of  preserving  an 
equable  chemical  susceptibility  in  the  surfaces  presented  to  the  light 
of  the  different  stars.  It  cannot  be  doubted  however  that  this  ele- 
ment can  be  kept  so  far  under  control  that  the  errors  introduced  will 
not  exceed  those  produced  by  atmospheric  perturbations  or  from 
other  disturbing  agencies  which  cannot  be  counteracted. 

»• 

That  this  latter  objection  has  been  met  is  shown  by  a  positive 

statement  made  by  Hartmann: l 

We  assume  here  only  that  every  plate  has  the  same  sensitiveness 
over  its  whole  surface,  and  that  the  development,  and  other  treat- 
ment of  the  plate,  have  been  precisely  the  same  for  all  different  points. 
If  we  should  not  make  these  two  assumptions,  the  photometric 
utilization  of  photographic  plates  would  be  entirely  impossible. 

The  next  point  to  be  considered  concerns  the  appearance 
and  formation  of  the  star  image.  Certain  facts  in  regard  to  it 
are  obvious  to  any  one  who  has  examined  many  photographs  of 
the  heavens;  namely,  the  fact  that  the  star  images  are  not 
round  over  the  entire  plate,  while  they  may  be  perfectly  so  near 
its  center.  Farther  away,  they  will  be  elongated,  usually  in  the 
direction  of  the  radius.  Their  density  is  also  irregular  away 
from  the  center,  so  that  the  discs  are  often  anything  but  uni- 
form. Sometimes  they  are  elongated  with  the  maximum  of 
density  at  one  end  of  the  ellipse,  making  them  quite  unsym- 
metrical  and  difficult  to  measure.  There  are  sometimes  spuri- 
ous images  on  a  plate,  false  stars,  which  however  can  usually 
be  distinguished  from  the  real  star  by  their  appearance.  It  is 
not  known  whether  they  occur  in  the  development  or  prepara- 
tion of  the  plate.  The  appearance  of  the  image  is  also  largely 
affected  by  the  accuracy  of  the  guiding  of  the  telescope,  for 
1  Ap.  J.t  10, 322. 


Plate  V 

DOUBLE-SLIDE  PLATE-CARRIER  ON  THE  40-INCH  TELESCOPE 
YERKES  OBSERVATORY 


PHOTOGRAPHIC  PHOTOMETRY  139 

which  two  devices  are  in  general  use  and  may  well  be  described 
here.  In  one  case  a  second  telescope  is  mounted  parallel  to  the 
photographic  instrument,  at  which  an  observer  is  placed  whose 
duty  is  to  get  the  instrument  accurately  pointed  by  setting  the 
crosswires  on  a  star,  and  then  to  keep  it  in  that  position  during 
the  entire  exposure.  With  a  large,  heavy  instrument  there  are 
usually  extra  motors,  which  may  be  used  or  not  at  the  will  of 
the  observer,  and  will  put  the  telescope  into  the  correct  position 
again  if  at  any  time  the  clock  work  should  fail  to  keep  it  true. 

The  other  device  for  guiding  is  an  integral  part  of  the  plate 
holder  which  is  attached  to  the  end  of  the  photographic  tele- 
scope. In  the  method  previously  described,  this  is  firmly  fas- 
tened to  the  tailpiece,  and  while  it  may  be  removed,  it  is  not 
movable.  In  the  present  case  the  plate  holder,  which  is  called 
a  double  slide  plate  carrier,  is  movable  itself,  and  slides  in  two 
directions  at  right  angles  to  each  other,  the  motion  being  easily 
controlled  by  two  screws.  The  main  telescope  is  carried  by  its 
clock  work,  which  must  be  accurate,  though  not  sufficiently  so 
for  the  fine  purposes  of  photographic  work.  A  star  on  the  edge 
of  the  field  is  used  as  a  guiding  star,  and  its  light  is  reflected  out 
to  the  side  and  into  the  eyepiece  by  means  of  a  totally  reflecting 
prism,  situated  just  within  the  framework  carrying  the  plate 
holder.  Here  the  observer  places  his  eye,  and  as  before  sets  the 
crosswires  on  the  star  selected  for  the  purpose,  and  keeps  his 
watch,  moving  the  plate  holder  as  the  necessity  arises.  This 
apparatus  has  been  used  very  successfully  with  a  large  tele- 
scope, such  as  the  Yerkes,1  where  the  use  of  a  guiding  telescope 
of  suitable  size  to  ensure  accurate  guiding  would  be  impossible. 
The  attachment  is  illustrated  in  the  accompanying  photograph, 
which  shows  the  end  of  the  forty-inch  with  the  plate  carrier  at- 
tached, the  plate  holder  being  removed.  On  two  sides  may  be 
seen  the  screws  which  control  the  motion,  and  in  the  upper  part 
the  guiding  eyepiece  projects  at  the  side,  while  the  totally 
reflecting  prism  is  dimly  visible  just  within  the  edge  of  the  box. 

The  next  point  to  be  considered  is  the  formation  of  the  star 
1  G.  W.  Ritchey,  Yerkes  Obs.  Pub.t  2,  389, 


140          THE  STUDY  OF  VARIABLE  STARS 

image.  A  star  is  a  point,  and  its  light  impinges  upon  only  the 
minute  spot  on  the  plate  which  corresponds  to  the  center  of 
the  image;  how  is  it  then  that  a  disc  of  regular  form  results  from 
the  chemical  action?  Charlier  *  answered  this  question  by  stat- 
ing that  the  light  of  the  star  falling  upon  the  plate  is  scattered 
either  by  fluorescence  or  reflection.  A  portion  is  thrown  to  the 
side  through  the  gelatine  film  and  by  its  chemical  action  pro- 
duces the  star  image,  while  the  rest  of  the  light  is  scattered. 
The  light  decomposes  the  silver  salt  which  is  spread  over  the 
plate  through  the  medium  of  the  gelatine  film;  the  action  of  the 
developer  causes  the  silver  particles  to  be  deposited  wherever 
the  light  has  fallen  upon  the  plate,  and  the  fixative  washes 
away  the  silver  salt  which  has  not  been  affected  by  the  light. 
The  star  image  consists,  then,  of  an  aggregate  of  silver  particles, 
and  hence  its  character  will  depend  upon  the  size  of  the  grains 
and  the  uniformity  of  their  distribution.  The  size  of  the  grain 
depends  upon  the  brand  of  plate,  being  largely  under  the  con- 
trol of  the  manufacturer.  Also  some  brands  contain  more  silver 
than  others  in  the  salt.  An  important  investigation  of  this  mat- 
ter was  made  at  the  Lick  Observatory  by  Perrine,2  who  exam- 
ined the  images  formed  on  several  kinds  of  plates  having  differ- 
ent lengths  of  exposure,  and  with  the  use  of  several  kinds  of 
developers.  He  found  that  the  best  results  could  be  obtained 
by  giving  the  light  time  to  act  entirely  through  the  thickness 
of  the  gelatine  film,  and  by  a  full  but  slow  development.  Thin 
films  appear  to  give  much  more  reliable  results  than  thick  ones, 
particularly  for  fast  work,  and  rapid  plates  with  an  increased 
proportion  of  silver  are  found  to  yield  more  accordant  results 
than  plates  with  the  normal  amount  of  silver.  Thus  it  will  be 
seen  that  the  formation  of  the  star  image  on  a  photographic 
plate  depends  upon  several  factors,  the  general  character  of 
the  plate  and  the  length  of  exposure  time  being  the  two  most 
prominent. 

The  appearance  of  the  star  image  is  also  affected  by  the 
instrument  with  which  it  is  photographed.  On  every  negative 
1  A.G.,  xix,  3.  2  L.O.B.,  nos.  143,  148. 


Plate  VI 

THE  TWO-FOOT  REFLECTOR,  YERKES  OBSERVATORY 


PHOTOGRAPHIC  PHOTOMETRY  141 

taken  with  a  reflecting  telescope  the  images  of  the  bright  stars 
will  have  rays  more  or  less  marked  extending  from  them,  usu- 
ally four  prominent  ones  at  right  angles  to  each  other,  and  four 
others  not  so  strong  half  way  between.  This  is  due  to  diffrac- 
tion around  the  supports  of  the  second  mirror,  which  is  placed 
near  the  opening  of  the  tube  in  order  to  reflect  the  light  out  to 
the  side  where  the  plate  holder  or  eyepiece  is  placed.  The  phe- 
nomenon does  not  occur  in  photographs  taken  with  a  refracting 
telescope. 

Sometimes  a  very  bright  star,  photographed  with  a  refractor, 
will  be  surrounded  with  a  ring  or  halo  of  light  some  little  dis- 
tance from  it.  The  halo  is  due  to  the  action  of  an  excess  of  light 
which  has  penetrated  entirely  through  the  film  to  the  back  of 
the  glass  plate  and  is  then  reflected  toward  the  gelatine,  upon 
which  it  acts  just  as  the  incident  light  does.  This  occurs  only 
with  a  bright  star,  and  cannot  be  avoided,  ordinarily.  It  comes 
about  necessarily  as  a  result  of  the  effort  to  make  the  exposure 
long  enough  to  get  the  faint  stars,  for  then  the  action  of  the 
bright  stars  goes  on  too  long. 

Another  effect,  dependent  upon  the  kind  of  telescope  used, 
has  to  do  with  chromatic  aberration,  which  is  quite  apparent 
with  the  refractor,  but  is  non-existent  with  the  reflector.  The 
object  glass  of  a  telescope  does  not  bring  all  of  the  colors  in  a 
beam  of  white  light  falling  upon  it  to  a  focus  at  the  same  dis- 
tance behind  the  object  glass.  It  can  be  ground  so  as  to  bring 
part  of  the  colors  together  in  the  same  focal  plane,  the  selection 
of  which  is  largely  under  the  control  of  the  manufacturer, 
and  is  made  to  depend  upon  the  purpose  for  which  the  telescope 
is  used.  If  it  is  to  be  used  for  visual  work,  then  the  colors  to 
which  the  eye  is  most  sensitive,  namely,  the  orange,  yellow, 
green  and  blue  must  be  brought  together.  In  this  case,  the 
focal  point  of  the  violet  and  ultra-violet  may  be  several  milli- 
meters farther  in,  the  distance  depending  upon  the  aperture  of 
the  lens  and  its  focal  length.  These  distances  for  different  wave- 
lengths must  be  found  on  a  scale  which  is  attached  somewhere 
at  the  eye  end  of  a  telescope,  and  the  investigation  of  this  mat- 


142          THE  STUDY  OF  VARIABLE  STARS 

ter  is  one  of  the  earliest  pieces  of  work  to  be  done  after  a  large 
telescope  has  been  set  up.  It  is  carried  out  by  taking  photo- 
graphs of  the  spectrum  at  different  focal  distances.  When  the 
values  have  been  found,  they  are  usually  plotted,  with  the 
wave-length  as  abscissa  and  the  distance  as  ordinate,  and 
the  resulting  curve  is  called  the  color  curve  of  the  telescope. 
The  following  diagram  shows  the  color  curve  of  the  Yerkes 
40-inch  telescope,  which  was  determined  by  Fox  l  in  1908. 

The  numbers  at  the  side  represent  the  change  in  focal  dis- 
tance in  mm.,  the  upper  part  of  the  figure  being  toward  the 
objective.  The  long  curve  has  its  flattest  part  from  X  5000  to 
X  6400,  showing  that  the  lens  is  corrected  for  visual  rays.  The 
small  curve  shows  the  effect  of  introducing  the  correcting  lens, 
which  is  used  for  photographing  stellar  spectra.  It  brings  the 
rays  from  X  4000  to  X  5000  to  a  focus  at  the  same  distance  be- 
hind the  object  glass,  thus  covering  the  region  used  for  the 
photographs.  A  further  reference  to  its  use  will  be  found  in 
the  chapter  on  spectroscopic  binaries. 

Since  the  photographic  plate  is  most  sensitive  to  the  action 
of  the  light  from  the  blue  and  violet  end  of  the  spectrum,  it  is 
evident  that  the  visual  telescope  will  not  ordinarily  take  good 
photographs,  but  that  an  instrument  must  be  specially  con- 
structed for  the  purpose.  There  are  certain  devices  which  can 
be  used  to  overcome  this  obstacle,  such  as  using  with  a  visual 
telescope  a  yellow  color  screen,2  which  will  cut  out  the  violet 
rays  that  are  out  of  focus  and  allow  a  sharp  image  to  be  formed. 
Sometimes  a  specially  stained  plate,  which  has  been  made  sen- 
sitive to  the  action  of  red  light,  can  be  used.  The  latter  method, 
which  is  of  great  importance,  will  be  treated  at  length  later  on. 

The  reflecting  telescope  is  entirely  free  from  this  defect,  and 
hence  all  of  its  light  is  utilized  in  forming  the  star  image,  while 
it  is  obvious  that  with  the  refractor  part  of  the  light  must  be 
lost,  whatever  the  method  employed. 

The  focal  length  of  the  telescope  has  an  important  bearing 
on  the  photographic  result.  The  objective  may  have  a  long  or 

1  Ap.  J.,  27,  252.  «  G.  W.  Ritchey,  Yerkes  Obs.  Pub.,  2,  389. 


MOOO 


A6000 


X7000 


mom. 
570 


550 
540 
530 
52,0 
510 
500 
490 
4-80 
470 
4GO 
450 


\ 


Figure  21 

COLOR  CURVE  OF  THE  40-INCH  OBJECTIVE,  YERKES  OBSERVATORY 


144          THE  STUDY  OF  VARIABLE  STARS 

a  short  focal  length,  depending  upon  the  purpose  to  which  it  is 
to  be  put.  A  large  lens  with  a  short  focus  will  give  a  very  bright 
image,  but  a  small  one;  while  another  of  the  same  aperture,  but 
greater  focal  length,  will  give  a  larger  image,  but  one  not  so 
bright.  The  greater  the  distance  of  the  focal  point  from  the 
lens  the  larger  the  image,  though  it  continually  grows  fainter 
per  unit  of  surface.  It  is  obvious  that  the  total  amount  of  light 
falling  upon  the  plate  will  be  the  same  in  both  cases,  but  where 
the  telescope  has  a  small  focal  length  this  light  will  be  con- 
densed into  a  small  image,  which  will  therefore  be  brighter  than 
the  larger  image  resulting  from  the  greater  focal  length.  This 
relation  is  usually  represented  numerically  by  the  ratio  between 
the  aperture  of  the  objective  and  its  focal  length;  e.g.,  the  ratio 
1 : 5  would  indicate  an  instrument  of  short  focus,  1 : 16  is  ordi- 
nary visual  length,  and  1 : 20  or  1 : 30  would  be  considered  a  long 
focus.  The  forty-inch  Yerkes  telescope  has  a  ratio  of  about 
1:19.  The  two-foot  reflector  has  a  ratio  of  about  1:4.  The 
Bruce  photographic  telescope,  as  shown  in  Plate  VII,  con- 
structed under  the  direction  of  Professor  Barnard,1  has  three 
telescopes  in  one  mounting;  two  photographic,  which  have  aper- 
tures of  ten  inches  and  six  and  one-fourth  inches,  and  focal 
lengths  of  50. 3  inches  and  thirty-one  inches,  being  both  in  the 
ratio  1:5;  and  the  third,  a  visual  telescope  for  guiding,  of  five 
inches  aperture  and  focal  length  apparently  the  same  as  the 
ten-inch  photographic  of  fifty  inches,  making  its  ratio  1:10. 
It  will  be  seen  that  the  short  focus  lenses  have  great  light- 
gathering  power,  which  makes  them  especially  suited  for  the 
photography  of  the  fainter  celestial  bodies,  while  the  longer 
focus  instrument  with  the  larger  image  is  better  for  visual  work 
in  which  micrometric  measurements  are  to  be  made. 

The  preceding  remarks  bear  upon  stellar  photography  in 
general,  but  we  may  now  pass  to  a  consideration  of  some  of  the 
points  which  have  to  do  directly  with  the  relation  between 
visual  and  photographic  magnitudes.  In  making  observations 
of  variable  stars  we  compare  the  brightnesses  of  different  points 
1  Ap.  J.,  21, 35. 


Plate  VII 

THE  BRUCE  PHOTOGRAPHIC  TELESCOPE,  YERKES  OBSERVATORY 


PHOTOGRAPHIC  PHOTOMETRY  145 

of  light  as  they  affect  the  retina,  while  in  making  photographic 
observations  we  must  compare  the  sizes  of  the  images  impressed 
upon  the  photographic  plate.  If  any  of  the  conditions  described 
above  alter  in  any  way  the  response  of  the  plate  to  the  relative 
brightness  of  the  stars  in  the  sky,  then  the  deductions  made  by 
measuring  the  sizes  of  the  star  images  will  be  erroneous.  Evi- 
dently color  is  a  most  important  element  to  be  considered.  It 
has  been  stated  that  the  blue  and  violet  rays  in  the  spectrum 
have  the  strongest  actinic  power,  and  hence  will  make  the 
strongest  impression  upon  the  photographic  plate.  The  stars 
which  are  strong  in  this  part  of  the  spectrum  will  therefore  make 
larger  images  on  the  plate  than  stars  which  are  deficient  in  it, 
even  though  of  apparently  the  same  brightness  to  the  eye.  The 
difference  between  the  two  impressions  will  depend  on  the 
spectral  type  of  the  star.  According  to  the  classification  of 
stellar  spectra,  stars  of  the  Sirian  type  are  strong  in  the  violet 
and  ultra-violet  and  are  very  white  in  color.  Stars  of  the  solar 
type  have  more  lines  in  this  part  of  the  spectrum  and  are  yel- 
lowish in  color,  a  condition  which  results  from  the  cutting  off 
of  a  part  of  the  violet  light  of  the  star.  Stars  of  Secchi's  types 
III  and  IV  have  large  general  absorption  in  the  violet  end  of 
the  spectrum  and  are  decidedly  red  in  color;  therefore  the  redder 
the  star  the  greater  the  difference  between  its  visual  and  photo- 
graphic magnitude  in  relation  to  white  stars.  While  for  pur- 
poses of  photometric  work  it  is  necessary  to  eliminate  this 
difference,  its  very  existence,  as  hinted  before,  has  led  to  its 
being  used  to  determine  the  spectral  type  of  stars  from  the 
difference  between  their  visual  and  photographic  magnitudes. 
Successful  efforts  have  been  made  to  overcome  the  difficulty 
by  staining  the  photographic  plate  so  as  to  make  it  more  sensi- 
tive to  the  long  waves.  But  before  describing  them,  we  must 
first  discuss  the  method  of  determining  the  magnitude  of  a  star 
from  the  measurements  of  its  photographic  image. 

In  1889  Charlier1  investigated  the  method  of  applying  stellar 
photography  to  the  determination  of  the  magnitudes  of  the 
1  A.G.,  xix,  1. 


146          THE  STUDY  OF  VARIABLE  STARS 

stars.  He  stated  the  problem  thus:  "To  determine  a  function 
which  shall  represent  the  relation  between  the  size  of  the  photo- 
graphic images  and  the  photographic  brightness,  in  which  the 
constants  shall  be  so  determined  that  the  resulting  photo- 
graphic brightnesses  shall  agree  on  an  average  with  those  ob- 
tained by  photometric  observations."  The  formula  already 
in  use  was 

(1)  m=a-61ogZ>, 

where  m  was  the  magnitude,  D  the  measured  diameter,  and  a 
and  6  constants.  &  is  a  number  which  depends  upon  the  instru- 
ment and  kind  of  plate  used,  while  a  depends  upon  the  clear- 
ness of  the  atmosphere  and  the  duration  of  the  exposure,  a, 
then,  will  vary  with  each  plate,  but  6  will  be  constant  so  long 
as  the  same  brand  of  plate  is  used  with  the  same  instrument. 
Charlier  tested l  the  formula  by  taking  photographs  of  the 
Pleiades  with  four  different  lengths  of  exposure,  13  m.,  l£  h., 
2  h.,  3  h.  He  then  selected  fifty-two  stars,  the  photometric  mag- 
nitudes of  which  had  been  very  well  determined  by  Lindemann, 
and  measured  their  diameters.  Each  star  afforded  an  equation 
of  the  form  (1),  there  being  fifty-two  in  all,  which  were  then 
solved  by  Cauchy's  method  for  a  and  b  with  the  following 
results:  — 

b  =  6.719 

=  6.779 

=  6.683 

=  6.814 

Mean  =  6.75, 

showing  b  to  be  constant  and  independent  of  the  time  of  expo- 
sure. In  order  to  find  a  he  transformed  equation  (1)  into 

a  =  m  +  6.75  log  Z>, 
with  the  resulting  values 

a=  18.77 
=  20.71 
=  20.89 
=  21.02, 

i  A.G.,  xix,  9. 


PHOTOGRAPHIC  PHOTOMETRY  147 

the  variation  depending  on  the  length  of  the  exposure.  In  order 
further  to  test  the  formula  he  substituted  values  for  a,  6,  and  D, 
in  equation  (1)  for  each  star,  and  compared  the  resulting  value 
of  ra  with  the  initial  value.  The  average  difference  photom.  — 
photog.  was  ±  .22  mg.  Among  the  residuals  the  value  0.6 
occurred  twice,  0.5  twice,  0.4  four  times,  0.3  twelve  times,  0.2 
twelve  times,  0.1  ten  times,  0.0  seven  times.  Many  other  points 
were  included  in  the  investigation,  which  was  carried  out  in  a 
thorough  manner,  and  the  formula  has  been  in  quite  general 
use  since.  However,  other  investigators  have  made  use  of  a 
somewhat  different  function  of  D.  Parkhurst,1  for  example, 
adopted  the  form 

Mag.=  a-6j/>, 

where  a  is  a  constant  for  each  plate,  depending  on  the  exposure, 
while  b  is  a  function  of  the  emulsion  and  conditions  of  develop- 
ment, which  are  kept  constant  in  agent,  time,  and  temperature. 
For  example,  using  a  Seed  plate  on  a  twenty-four  inch  reflector, 
he  found  on  developing  it  ten  minutes  in  hydro-quinone  at  20° 
C.  that  the  value  of  b  was  .94,  the  unit  being  .001  mm.,  and  this 
was  constant  so  long  as  the  above  conditions  were  observed. 
The  value  of  a  was  found  for  each  plate  by  using  visual  magni- 
tudes of  white  stars.  At  another  time  Parkhurst 2  found  the 
formula 

m=a-D™ 
to  fit  the  Cramer  plates  better. 

Whichever  formula  is  adopted  the  net  result  is  the  same. 
From  the  magnitudes  of  the  known  stars  on  the  plate  and  the 
measured  diameters  of  their  photographic  images  the  constants 
a  and  b  can  be  determined,  and  thereafter  used  in  finding  the 
magnitudes  of  the  unknown  stars.  For  a  given  series  of  plates 
6  need  be  determined  but  once,  for  it  is  constant,  while  a  de- 
pends on  the  length  of  exposure,  which  is  different  for  each 
plate.  In  determining  both  a  and  b  white  stars  should  be  used. 
Hence  a  knowledge  of  their  spectra  is  essential  before  the  stand- 
ard stars  can  be  selected. 

1  Ap. «/.,  27, 171.  *  Ap.  J.,  23,  79. 


148          THE  STUDY  OF  VARIABLE  STARS 

We  are  now  ready  to  consider  some  of  the  applications  which 
have  been  made  of  this  method  of  determining  photographic 
images.  Obviously  it  can  be  used  in  the  study  of  variable  stars, 
particularly  of  the  short  period  variables.  Sometimes  on  the 
same  plate  at  regular  intervals  of  time  several  exposures  are 
taken,  from  which  the  variation  in  brightness  can  be  deter- 
mined. So  many  researches  of  this  sort  have  been  made  that 
it  is  not  possible  to  mention  them  all.  Some  interesting  anoma- 
lies have  appeared  as  a  result,  though  they  still  lack  absolute 
confirmation.  The  light  curves,  as  determined  by  photometric 
and  photographic  observations,  do  not  always  agree.  Some- 
times the  form  is  different  and  sometimes  the  time  of  minimum 
is  not  the  same. 

Reference  has  been  made  at  several  points  to  the  necessity 
of  using  white  stars  as  standards.  Nevertheless  the  existence 
of  red  stars  cannot  be  ignored,  and  it  is  a  persistent  fact  that  a 
red  star  will  not  give  an  image  on  a  photographic  plate  which 
will  be  a  measure  of  its  visual  brightness;  hence  some  method 
of  correcting  for  this  difference  must  be  found,  or  else  red  stars 
cannot  be  studied  photographically.  Experiments  were  con- 
ducted at  the  Yerkes  Observatory  to  discover  if  plates  could 
not  be  stained  with  some  dye  which  would  make  them  sensitive 
to  the  visual  maximum  of  the  spectrum,  which  extends  from 
X  5000  to  5900.  Many  such  dyes  had  already  been  investigated 
elsewhere,  but  the  results  did  not  seem  to  be  entirely  satisfac- 
tory; hence  the  need  for  a  further  effort.  The  work  was  placed 
in  the  hands  of  Wallace,1  who  experimented  assiduously  with 
several  different  kinds  of  dyes  until  he  found  one  which  gave 
the  desired  result,  from  a  combination  of  pinacyanol  +  pina- 
verdol  +  homocol.  The  resulting  plate  he  called  Pan-iso.  Since 
it  was  still  a  little  defective,  he  prepared  a  compensation  filter, 
and  the  two  together  produced  the  desired  effect. 

The  new  stained  plates  were  then  used  by  Parkhurst  and 
Jordan,2  first  to  show  that  visual  magnitudes  of  red  stars  could 
be  obtained  photographically,  by  taking  plates  of  stars  such 
1  Ap.  J.,  26,  299.  J  Ap.  J.t  27, 169. 


PHOTOGRAPHIC  PHOTOMETRY  149 

as  U  Cygni;  and  secondly  in  carrying  out  a  research  entitled 
The  Photographic  Determination  of  Star  Colors  and  their  Relation 
to  Spectral  Type.  The  purpose  of  this  important  work  cannot 
be  better  stated  than  in  the  authors'  own  words  :  — 

It  has  long  been  recognized  that  eye  estimates  form  a  very  unsatis- 
factory method  of  determining  star  colors,  and  an  urgent  need  has 
been  felt  for  some  means  of  accurate  measurement.  The  plan  we  are 
following  seems  to  supply  that  need,  and  also  aids  in  the  solution  of 
two  very  interesting  problems.  First  it  enables  us  to  co-ordinate  visual 
and  photographic  magnitudes,  thus  allowing  us  to  use  as  standards 
the  visual  magnitudes  of  the  white  stars  from  the  best  modern  photo- 
metric catalogues,  and  at  the  same  time  avoid  many  of  the  inherent 
difficulties  and  systematic  errors  of  visual  measures  of  colored  stars. 
Second,  important  data  are  added  for  the  study  of  stellar  evolution 
since  the  relation  of  color  to  the  stages  of  stellar  development  is  very 
close  and  capable  of  quite  precise  determination. 

Our  method  is  based  on  a  suggestion  first  made  (as  far  as  we  are 
aware)  by  Schwartzschild,  that  the  difference  between  the  visual  mag- 
nitude of  a  star  and  that  obtained  from  ordinary  photographic  plates 
would  give  an  accurate  measure  of  the  star's  color.  He  called  this 
difference  "Farbentonung,"  or  color  index.  Our  addition  consists  in 
determining  the  visual  magnitudes  also  by  photographic  means,  and 
making  both  determinations  practically  simultaneous.  With  this  in 
view,  pairs  of  ordinary  and  iso-chromatic  plates  were  taken  regularly 
with  the  24  inch  reflecting  telescope  of  this  Observatory,  and  a 
method  was  suggested  of  deriving  the  "visual"  magnitude  from  the 
iso-chromatic  plates. 

The  two  kinds  of  plates  used  in  this  work  were  Seed  27,  the 
"ordinary"  brand,  and  Wallace's  Pan-iso  plates.  400  color 
pairs  were  obtained,  and  the  photometric  magnitudes  obtained 
by  the  formula 


The  results  were  correlated  in  several  different  ways:  first,  the 
difference  in  magnitude  between  Seed  and  Pan-iso,  when 
formed,  was  called  the  color  intensity;  secondly,  the  magnitudes 
of  thirty  stars,  obtained  from  Pan-iso  plates,  were  compared 
with  the  Potsdam  magnitudes  obtained  photometrically,  the 
agreement  showing  by  the  small  differences  that  visual  magni- 
tudes could  actually  be  obtained  from  photographic  plates; 


150 


THE  STUDY  OF  VARIABLE  STARS 


thirdly,  a  list  of  forty-nine  stars  was  arranged  in  order  of  color 
intensity  and  tabulated,  the  spectra  showing  that  as  the  stars 
advanced  in  type,  the  color  intensity  became  greater;  e.g.,  two 
V  type  stars  at  the  beginning  of  the  list  had  color  intensity  .02 
and  .03,  and  two  K-M  stars  at  the  end  had  1.83  and  1.86. 
The  two  sets  of  values  in  the  table  were  plotted,  the  spectral 
types  being  laid  off  as  ordinates,  and  the  differences  in  magni- 
tude, visual  —  photographic,  as  abscissas,  and  the  accompany- 
ing curve  was  drawn. 

0M0  l"0  rfo 


M 


Figure  22 

SPECTRAL  TYPE  AND  COLOR  INTENSITY 

The  same  curve  can  be  used  to  obtain  the  spectral  type  for 
faint  stars,  since  the  difference  vis.  —  photog.  magnitude  can 
be  obtained  by  comparing  the  results  from  the  two  sets  of 
plates,  and  the  resulting  A  m,  or  color  intensity,  when  used  as 
an  abscissa  with  the  curve,  will  give  for  the  ordinate  the  spec- 
tral type. 


PHOTOGRAPHIC  PHOTOMETRY  151 

A  similar  curve  was  obtained  by  King,1  working  at  Harvard 
Observatory,  in  a  different  manner.  He  obtained  photographic 
magnitudes  of  109  bright  stars  by  measuring  the  densities  of 
their  extra-focal  images,  in  a  manner  presently  to  be  described, 
and  compared  them  with  photometric  measures  from  the 
Revised  HP.  The  differences  were  then  arranged  in  order  of 
spectral  type,  Oe  —  M.  The  means  of  the  values  of  A  m  were 
taken,  and  the  results  plotted  as  abscissas  with  the  spectral 
type  as  ordinate,  resulting  in  a  curve  very  similar  to  that  of 
Parkhurst. 

Reference  has  just  been  made  to  the  method  of  obtaining 
photographic  magnitudes  by  means  of  measuring  the  density 
of  extra  focal  images.  As  the  process  is  somewhat  complicated 
a  rather  detailed  explanation  is  necessary.  The  extra  focal 
images  must  be  compared  with  a  series  of  standard  images,  the 
densities  of  which  have  been  measured,  and  correspond  to 
known  differences  in  magnitude.  The  two  sets  of  images  cannot 
be  compared  directly,  but  through  the  medium  of  a  photo- 
graphic wedge,  which  may  be  prepared  in  several  different 
ways,  the  only  requisite  being  that  the  increase  of  blackening 
must  be  uniform.  The  method  of  preparing  the  wedge  at 
Harvard 2  is  to  expose  a  plate  to  light  entering  through  a  tri- 
angular-shaped opening  bounded  by  logarithmic  curves  instead 
of  straight  lines.  Most  of  the  photographic  wedges  in  this 
country  have  come  from  this  Observatory.  The  wedge  and 
the  standard  scale  are  first  placed  in  the  measuring  machine, 
and  readings  taken  for  the  different  standard  images  on  it. 
Then  the  scale  is  removed,  the  star  image  brought  into  the 
field,  and  the  wedge  moved  until  the  two  densities  match  and 
the  reading  is  taken.  From  these  readings  the  differences  in 
magnitude  may  be  obtained. 

At  the  Harvard  Observatory3  the  custom  is  as  follows.  Each 
plate  is  capable  of  receiving  forty  images.  The  usual  order  is  to 

1  Annals,  H.C.O.,  vol.  59,  nos.  4,  5. 

8  E.  S.  King,  Annals,  H.C.O.,  41,  237,  and  59,  36. 

9  E.  S.  King,  Annals,  H.C.O.,  59,  95. 


152          THE  STUDY  OF  VARIABLE  STARS 

expose  first  on  Polaris  once  or  twice,  and  then  to  take  a  number 
of  stars,  at  foci  so  chosen  as  to  form  images  of  nearly  the  same 
density  as  Polaris.  Next  some  bright  star,  as  a  Lyrae,  at  each 
of  the  different  foci  is  taken,  in  order  to  obtain  a  scale  of  magni- 
tudes; and  finally,  Polaris  is  again  taken.  As  a  result  the 
standards  and  the  star  to  be  measured  are  on  the  same  plate, 
and  are  subject  to  exactly  the  same  conditions  of  exposure  and 
development. 

At  Yerkes  Parkhurst  and  Jordan1  used  a  set  of  standard 
magnitudes,  obtained  by  illuminating  squares  in  a  sensitometer 
box  by  light  passing  through  holes  of  different  diameters.  These 
magnitudes  were  investigated  and  a  table  formed  which  gave 
the  value  A  m  for  each  degree  of  blackening  compared  with 
that  from  the  first  hole  taken  as  a  standard.  The  values  of  A  m 
were  made  the  abscissas  for  the  curve  of  the  wedge,  and  the 
ordinates  were  the  scale  readings  corresponding  to  different 
thicknesses  of  the  wedge.  The  measurements  were  taken  with 
the  Hartmann  photo-micrometer.  This  method  has  been  used 
especially  in  measurement  of  light  curves  of  short  period  vari- 
ables. At  Harvard  the  method  has  been  employed  for  deter- 
mining the  magnitudes  of  bright  stars. 

Still  another  method  of  determining  stellar  magnitude  by 
means  of  photographic  images  has  been  used  successfully  at 
Harvard  for  many  years.  A  series  of  photographic  images  is 
prepared  which  represent  certain  differences  of  magnitude. 
The  unknown  stars  are  then  compared  with  the  scale,  the  differ- 
ences being  estimated  to  tenths.  The  initial  point  of  the  scale 
is  then  found  by  referring  to  the  magnitude  of  some  known 
star  on  the  plate.  The  scale  plate  at  Harvard 2  was  prepared 
by  taking  a  plate  of  the  Hyades,  and  giving  it  successive  expo- 
sures of  3,  9,  27,  81,  243,  729  seconds,  the  telescope  being 
moved  between  each  two  exposures.  Each  star  image  thus 
received  about  three  times  as  much  light  as  the  one  preceding 
it,  in  order  to  make  the  series  represent  differences  in  brightness 
equivalent  to  one  magnitude.  This  ratio  was  adopted  rather 
1  Ap.  J.,  26,  245.  2  Annals,  H.C.O.,  18, 120. 


PHOTOGRAPHIC  PHOTOMETRY  153 

than  2.5,  as  experiments  showed  the  latter  ratio  to  be  too  small. 
From  the  photograph  thus  taken  the  strip  of  images  of  one 
star  was  cut  out  and  mounted  in  a  suitable  frame.  It  was  pro- 
tected by  cementing  a  piece  of  thin  cover-glass  over  it,  and  a 
handle  served  to  hold  it  over  the  image  to  be  measured  in  such 
a  way  that  the  comparison  was  readily  made. 

This  method  has  been  applied  to  a  study  of  variable  stars. 
Sequences  of  comparison  stars  have  been  selected,  and  their 
magnitudes  determined  by  comparing  them  with  the  standard 
plate.  The  magnitudes  of  the  variables  have  then  been  deter- 
mined by  Argelander's  method  of  estimation. 

It  would  be  impossible  in  the  brief  space  allowed  to  this  chap- 
ter even  to  mention  all  that  has  been  done  in  photographic 
photometry,  but  it  is  hoped  that  the  main  points  have  been 
covered.  An  excellent  resume*  of  the  subject  is  to  be  found  in  a 
brief  paper  by  Pickering,  H.C.O.,  Annals,  vol.  71,  no.  1. 

The  suggestion  has  also  been  made  that  in  order  to  eliminate 
any  errors  arising  from  imperfections  in  the  eye,  which  still 
must  be  used  in  making  comparisons  even  with  the  photo- 
micrometer,  it  might  be  possible  to  introduce  some  automatic 
device  which  would  replace  the  eye,  such  as  a  galvanometer, 
and  use  it  in  measuring  the  density  of  the  star  image.  In  Har- 
vard Circular,  no.  155,  Pickering  considers  the  possibility  of 
such  devices,  and  suggests  several,  such  as  the  thermal-pile, 
bolometer,  radiometer,  or  selenium  cell.  The  only  duty  left 
for  the  eye  to  perform  would  be  to  read  the  deviation  of  the 
indicator.  These  methods  have  not  as  yet  been  applied  to  the 
measurement  of  photographic  images,  though  the  selenium  cell 
has  been  very  successfully  used  by  Stebbins  in  observing  with 
the  telescope.  A  description  of  his  work  will  form  part  of  the 
next  chapter. 


CHAPTER  VIII 
PHOTO-ELECTRIC  PHOTOMETRY 

Two  more  types  of  photometer  have  been  introduced  lately 
which  bid  fair  to  supersede  the  older  ones,  particularly  for  stars 
where  the  fluctuations  in  light  are  small,  yet  rapid.  In  both 
types  the  light  from  the  star  is  received  on  a  surface  which  is 
electrically  sensitive,  so  that  a  change  in  the  intensity  of  the 
light  falling  upon  it  causes  a  change  in  the  resistance,  or  poten- 
tial, which  is  registered  by  some  electrical  apparatus.  The  two 
types  in  use  (so  far  as  the  writer  knows)  are  the  selenium  cell, 
successfully  adapted  by  Stebbins 1  at  the  University  of  Illinois 
for  stellar  photometry,  and  the  photo-electric  cell,  which  has 
been  studied  for  some  time  in  physical  laboratories,  but  has 
only  recently  been  applied  to  problems  of  astronomical  photom- 
etry, among  others  by  Guthnick2  at  the  Observatory  of  Ber- 
lin, Meyer  and  Rosenberg3  at  Tubingen,  and  Schultz 4  at  the 
University  of  Illinois.  The  material  in  this  chapter  is  taken 
from  papers  published  by  these  men. 

The  great  value  of  these  photometers  lies  in  the  fact  that 
they  are  sensitive  to  extremely  small  variations  in  brightness. 
With  both  cells  the  difference  in  stellar  brightness  can  be 
measured  with  an  error  of  ±  .006  for  a  normal  magnitude.  The 
selenium  cell  will  be  described  first. 

The  physical  principle  on  which  it  is  based  may  briefly  be 
stated  as  follows.  The  crystalline  form  of  selenium  changes  its 
electrical  resistance  when  exposed  to  light,  or  under  certain 
circumstances  it  gives  an  electro-motive  force  when  illuminated, 
hence  this  form,  with  electrodes  attached,  was  early  called  a 
selenium  cell.  The  theory  of  its  application  to  stellar  photom- 

1  Ap.  J.,  27, 183;  32, 185;  26,  326;  39,  459;  34, 112.  Pop.  Ast.,  19,  1. 

2  Veroff.  der  K.  Sternwarte  zu  Berlin-Babelsberg,  Band  I.,  Heft  1. 
8  V.J.S.,  48,  210.  «  Ap.  J.,  38,  187. 


PHOTO-ELECTRIC  PHOTOMETRY  155 

etry  is  very  simple.  The  method  proposed  is  to  attach  the 
selenium  cell,  in  a  closed  case,  to  the  end  of  the  telescope,  expose 
its  surface  to  the  light  of  a  star,  and  note  the  change  of  resist- 
ance by  means  of  a  galvanometer.  The  great  objection  to  its 
employment  is  that  the  resistance  of  selenium  is  affected  by 
other  agencies  than  light,  and  is  difficult  to  handle  on  that 
account. 

The  cell  used  by  Stebbins  is  made  by  winding  wires  in  a 
spiral  form  about  a  flat  surface,  about  50  x  26  mm.  in  area, 
and  filling  the  space  on  one  side  with  selenium  which  has  been 
made  sensitive.  The  best  process  of  sensitizing  it  is  a  commer- 
cial secret.  The  wires  pass  out  at  the  back  of  the  surface  and 
are  connected  with  the  measuring  apparatus.  After  many 
experiments,  it  was  found  that  the  following  precautions  in  the 
use  of  the  cell  were  necessary.  First,  the  selenium  should  be 
kept  at  a  uniform  low  temperature,  0°  C.,  or  lower.  Second, 
the  current  should  pass  continually  through  the  selenium. 
Third,  exposures  to  light  should  be  short,  as  ten  seconds,  with 
longer  intervals  for  recovery.  In  consequence  of  the  first 
requirement  the  cell  is  placed  in  an  ice  chamber,  which  is  then 
attached  to  the  twelve-inch  telescope.  In  warm  weather  the  ice 
is  renewed  every  day,  but  in  winter  this  is  not  necessary.  A 
small  shutter  may  be  opened  in  order  to  expose  the  cell  to  star- 
light. The  current  is  supplied  by  a  few  dry  cells,  giving  an 
E.M.F.  of  six  volts.  For  best  results  it  must  be  applied  stead- 
ily, as  it  has  been  found  that  if  a  selenium  cell  of  3,000,000  ohms 
is  used  the  resistance  decreased  slowly,  until,  after  half  an  hour 
had  elapsed,  a  steady  condition  is  reached.  This  requires  that 
the  current  should  be  started  at  least  half  an  hour  before  obser- 
vations can  be  begun.  Selenium  also  requires  time  for  recovery 
from  light  action.  Hence  if  it  has  been  exposed  ten  seconds  to 
light  from  a  star,  a  wait  of  about  a  minute  is  necessary  in  order 
to  allow  it  to  regain  its  sensibility. 

As  arranged  for  measurement,  the  selenium  cell  is  made  one 
arm  of  a  Wheatstone  bridge.  Two  of  its  other  arms  are  con- 
stant, and  the  fourth  can  be  varied  to  produce  a  balance  with 


156          THE  STUDY  OF  VARIABLE  STARS 

the  cell,  the  resistance  of  which  may  change  from  night  to 
night.  The  deflection  is  read  by  a  d'Arsonval  galvanometer, 
which  is  at  rest  when  the  four  arms  are  in  balance.  The  zero 
point  does  not  remain  stationary  during  the  evening,  owing  to 
small  temperature  changes  in  the  selenium  cell.  Hence,  on 
beginning  a  night's  work  a  series  of  readings  is  taken,  which  is 
repeated  once  an  hour  in  order  to  determine  its  rate  of  change, 
and  the  drift  is  interpolated  for  the  time  when  the  stars  are 
measured.  In  practice  the  galvanometer  and  Wheatstone 
bridge  are  set  up  in  a  room  some  distance  away  from  the  dome, 
and  here  the  observer  is  stationed.  The  assistant,  who  is  at 
the  telescope,  makes  the  exposures  by  moving  the  shutter.  In 
this  case  as  with  all  other  photometers,  comparison  stars  are 
necessary.  An  observation,  or  "set,"  consists  of  opening  the 
shutter  of  the  cell  for  ten  seconds  in  the  order,  four  times  on  a 
comparison  star,  eight  times  on  the  variable,  four  times  on  the 
comparison  star  again,  making  sixteen  deflections  in  all.  As 
each  exposure  takes  ten  seconds,  and  the  recovery  one  minute, 
the  entire  "set"  requires  about  twenty  minutes.  In  observing 
Algol  from  four  to  six  sets  were  considered  sufficient,  unless  it 
was  near  the  time  of  minimum,  either  that  of  the  principal,  or 
of  the  expected  secondary  minimum. 

Extra  focal  images  of  the  stars  are  used  as  large  as  7  mm.  in 
diameter.  "Other  experiments  have  shown  that  for  faint 
sources  the  galvanometer  deflections  are  sensibly  proportional 
to  the  light  intensities,  and  therefore  the  ratio  of  the  deflections 
(with  Pogson's  rule)  gives  at  once  the  difference  of  magnitudes 
of  the  two  stars."  The  following  example  will  illustrate  this. 
On  January  7,  1910,  comparisons  were  made  of  a  Persei  and 
Algol,  with  the  following  results:  — 

A  =  a  Persei,  deflection  7.46  scale  divisions, 
B  =  £  Persei,  deflection  6.33     " 

then  |-p*-,  ' : 

log  7.46 -log  6.33      0.072 


PHOTO-ELECTRIC  PHOTOMETRY  157 

The  probable  error  of  a  normal  magnitude  near  the  principal 
minimum  was  ±  .023,  and  near  the  secondary  minimum, 
±  .006,  thus  showing  a  better  accordance  than  any  kind  of 
visual  or  photographic  work.  Corrections  were  made  for  the 
drift  of  the  galvanometer  zero  and  differential  atmospheric 
absorption.  While  this  method  seems  extremely  simple  in  the- 
ory, in  execution  it  is  not  so  easy,  since  it  requires  no  small 
amount  of  skill  in  manipulation. 

The  extreme  sensitiveness  of  the  cell  has  made  it  of  great 
value,  particularly  in  the  observation  of  short  period  variables. 
The  first  and  most  striking  result  obtained  by  it  was  the  dis- 
covery of  the  secondary  minimum  of  Algol.  This  star  is  a  well- 
known  eclipsing  binary,  in  which  one  component  is  bright,  and 
the  other,  because  of  the  absence  of  a  secondary  minimum,  has 
long  been  called  dark,  though  known  to  be  of  approximately 
the  same  size  as  the  primary.  From  the  first  it  seemed  a  diffi- 
cult fact  to  accept,  that-in  developing  from  the  same  primordial 
nebulous  mass,  one  star  should  be  bright  and  the  other  dark, 
and  the  eclipse  theory  was  rejected  by  many  astronomers. 
Later  spectroscopic  observations,  by  proving  the  binary  char- 
acter of  the  system,  seemed  to  present  further  proof  that  the 
companion  was  in  reality  dark,  for  there  was  only  one  set  of 
lines  in  the  spectrum.  The  greatly  increased  sensitiveness  of 
the  selenium  photometer  over  that  of  other  kinds  previously 
known  offered  an  opportunity  for  testing  the  light  of  Algol  at 
the  time  of  the  expected  secondary  minimum,  and  evidence  of 
it  was  indubitable,  as  the  curve  in  the  first  chapter  shows.  The 
variation  in  brightness  amounted  to  .06  mg. 

Other  very  interesting  results  have  been  published  recently 
by  Stebbins,  on  several  spectroscopic  binaries,  the  purpose 
being  to  discover  if  any  of  them  are  variables.  Of  the  eleven 
stars  examined,  four  were  proved  to  show  variation  in  bright- 
ness. Unfortunately,  work  with  this  photometer,  though  very 
accurate,  is  also  slow,  because  the  requirements  are  extremely 
exacting.  When  the  comparison  star  is  some  distance  from  the 
variable,  and  it  is  desired  to  secure  results  correct  to  .01  mg.,  it 


158          THE  STUDY  OF  VARIABLE  STARS 

is  useless  to  work  on  any  but  first  class  nights.  So  far  Stebbins 
has  not  tried  it  on  any  stars  fainter  than  third  magnitude.  In 
referring  to  other  work  he  mentions  the  use  of  a  potassium 
photo-electric  cell  by  Schultz,  and  adds  that  such  cells  have 
been  successfully  used  elsewhere. 

The  study  of  photo-electric  cells  is  of  such  recent  develop- 
ment that  the  subject  is  still  one  of  a  research  character,  and 
difficult  to  make  clear  in  a  simple  way.  The  author  will  not 
attempt  to  give  more  than  a  superficial  sketch  of  the  funda- 
mental principles,  and  then  proceed  to  describe  the  apparatus 
in  use  at  Berlin-Babelsberg.  The  theoretical  exposition  is  taken 
largely  from  a  recent  volume  of  Allen1  on  Photo-Electricity,  and 
only  points  which  bear  directly  on  stellar  photometry  will  be 
mentioned.  Some  of  the  material  can  best  be  expressed  in 
Allen's  own  words:  — 

The  term  photo-electricity  is  used  in  a  general  sense  to  designate 
any  electrical  effect  due  to  the  influence  of  light.  Thus  the  change  of 
electrical  resistance  of  selenium  when  exposed  to  light  is  spoken  of  as  a 
photo-electric  action.  The  term  is  more  particularly  used  to  denote  a 
change  in  the  electrification  of  a  body,  due  to  the  action  of  light.  In 
accordance  with  modern  electrical  theory,  light  is  an  electro-magnetic 
disturbance,  and  any  change  in  the  electrification  of  a  body  is  caused 
by  the  addition  or  removal  of  negative  electrons.  Hence,  from  this 
standpoint,  a  photo-electrical  change  is  equivalent  to  the  liberation  of 
negative  electrons,  under  the  influence  of  electro-magnetic  waves. 

The  first  experiment  showing  this  phenomenon  was  made  in 
1887,  by  Hertz,  who  noticed  that  when  ultra-violet  light  fell 
upon  the  spark-gap,  the  electrical  discharge  took  place  more 
easily  than  when  the  gap  was  not  illuminated,  and  the  greater 
the  actinic  power  of  the  source  of  light,  the  more  powerful  the 
effect.  In  the  following  year  it  was  shown  that  the  action  had 
its  seat  at  the  cathode,  or  negative  terminal  of  the  spark-gap. 
In  1889  experiments  by  Elster  and  Geitel  showed  that  electro- 
positive bodies,  like  sodium  and  potassium,  manifested  photo- 
electrical  activity  when  exposed  to  ordinary  light.  Freshly 
polished  surfaces  of  zinc  and  aluminum  exhibit  photo-electri- 
1  Photo- Electricity,  the  Liberation  of  Electrons  by  Light. 


PHOTO-ELECTRIC  PHOTOMETRY  159 

cal  effect  when  exposed  to  sunlight,  but  when  they  are  allowed 
to  stand  in  air  their  activity  rapidly  diminishes.  This  is  known 
as  "fatigue."  Later  Elster  and  Geitel  described  a  cell  in  which 
the  sensitive  surface  is  potassium,  placed  in  an  atmosphere 
of  argon  or  helium,  to  secure  permanence.  It  was  designed 
for  the  measurement  of  sunlight:  — 

Within  the  last  decade  great  progress  has  been  made  by  carrying 
out  experiments  in  a  high  vacuum,  where  conditions  are  much  simpli- 
fied through  the  absence  of  a  surrounding  atmosphere.  Two  principal 
methods  of  experiment  may  be  employed  [the  second  one  of  which 
applies  particularly  in  this  instance,  and  is  the  only  one  which  will  be 
referred  to]. 

The  current  flowing  between  the  illuminated  plate  and  a  parallel 
plate  may  be  measured  by  a  galvanometer  or  electrometer  when  a 
known  difference  of  potential  is  maintained  between  the  two  plates. 
.  .  .  The  second  method  measures  the  number  of  electrons  leaving  the 
illuminated  surface,  and  by  varying  the  potential  difference  applied  it 
is  possible  to  find  how  the  number  depends  on  the  strength  of  the  elec- 
tric field.  ...  If  an  accelerating  field  is  applied  the  number  leaving 
the  plate  will  rise  to  a  maximum  value,  so  that  for  further  increases  in 
the  potential  the  current  becomes  approximately  constant. 

Experiments  in  a  vacuum  in  which  the  intensity  of  the  incident  light 
was  varied  led  to  the  important  conclusions  that  .  .  .  (2),  the  num- 
ber of  electrons  emitted  is  directly  proportional  to  the  intensity  of  the 
light. 

The  practical  application  of  this  principle  may  best  be 
understood  by  reference  to  the  photometer  of  the  Neu  Babels- 
berg  Observatory,  which  is  depicted  in  Figure  23.  In  the  dia- 
gram KK  is  the  cell  chamber,  and  M  the  cell.  The  potassium 
film  forms  the  cathode  terminal,  "Kalium-Kathode,"  which  is 
connected  with  the  battery  cells,  which  maintain  it  at  a  con- 
stant potential.  The  other  terminal,  the  "Platin- Anode,"  is  a 
wire  of  platinum,  one  end  of  which  is  bent  iato  an  open  ring, 
and  placed  immediately  over  the  cathode,  while  the  other  end 
passes  out  of  the  cell  and  is  connected  directly  with  the  elec- 
trometer. Thus  an  electric  field  is  formed  in  which  a  constant 
difference  of  potential  is  maintained. 


160          THE  STUDY  OF  VARIABLE  STARS 

The  potassium  surface  thus  corresponds  to  the  illuminated 
plate,  and  the  "Platin-Anode"  to  the  parallel  plate.  The  light 
from  the  star,  in  the  form  of  an  extra-focal  image,  falls  upon 
the  potassium  surface  and  liberates  electrons  which  escape 
from  it,  are  caught  on  the  platinum  ring,  and  give  up  their 
charge,  which  is  conveyed  along  the  wire  to  the  electrom- 
eter. 

Since  in  a  vacuum  the  number  of  electrons  emitted  is  propor- 
tional to  the  intensity  of  the  incident  light,  it  follows  that  by 
means  of  the  currents  measured  by  the  electrometer  we  can 
determine  the  relative  intensities  of  the  light  from  two  stars, 
and  hence  find  their  difference  in  magnitude. 

This  brief  statement  shows  theoretically  the  action  taking 
place  in  a  photo-electric  cell.  With  it  as  a  preface  we  can  more 
easily  understand  the  principal  parts  of  the  apparatus  of 
Guthnick  and  Prager,  which  will  now  be  described,  as  far  as 
possible  in  their  own  words.  A  study  of  the  diagram  will  greatly 
aid  in  the  understanding  of  the  instrument,  which  is  attached 
to  a  30  cm.  refractor  of  5.1  meter  focal  length. 

The  apparatus,  as  used  with  the  telescope,  consists  of  four 
parts:  the  cell  chamber  with  the  cell,  the  electrometer,  the 
batteries  for  supplying  the  current  for  the  cell  and  the  electrom- 
eter, and  the  part  used  in  setting  on  the  star.  The  first  three 
are  the  necessary  equipment  of  any  photo-electric  cell,  and  the 
last  is  needed  only  for  stellar  work.  Several  accessories  are 
mentioned  which  are  required  for  testing  various  parts  of  the 
instrument  and  other  purposes. 

In  the  diagram  AA  is  the  ocular  end  of  the  telescope.  The 
focal  point  is  in  the  plane  BB,  where  an  iris  diaphragm  permits 
the  light  of  the  star  to  pass  through,  but  shuts  out  all  extrane- 
ous and  disturbing  light,  such  as  that  from  other  stars,  or  from 
the  heavens  when  brightly  lighted  by  the  moon.  The  cell  is 
about  20  cm.  farther  out,  and  hence  the  light  falling  on  it  is 
from  an  extra-focal  image  of  the  star. 

This  is  in  order  to  diminish  the  disturbing  effects  of  specks  of 
dust  which  will  collect  on  the  outer  walls  of  the  cell  and  the 


I  I  I  I  I  I! I  I  I  I !| I  I  I  I  I  I  I  I  I  I  I  I 


I  |  I  I  I  I  |  II  I  I  |  I  Ml  |  II 


Figure  23 

DIAGRAM  OF  THE  PHOTO-ELECTRIC  APPARATUS,  BERLIN-BABELSBERG 


162          THE  STUDY  OF  VARIABLE  STARS 

glass  window,  H,  of  the  cell  chamber.  As  far  as  the  measure- 
ments are  concerned,  it  is  immaterial  in  which  part  of  the  cone 
of  rays  the  cell  is  placed,  so  long  as  all  of  the  light  from  the 
objective  reaches  it. 

The  cone  of  rays  falls  upon  the  right-angle  prism,  C,  and  is 
reflected  out  into  the  small  telescope,  D.  Looking  into  the  eye- 
piece, the  observer  sees  the  opening  of  the  iris  diaphragm,  with 
the  star  in  its  center.  The  opening  ranges  from  1'  to  2'  in  diam- 
eter, in  order  to  prevent  the  violet  rays  from  being  cut  off  by 
its  edges.  With  a  reflector,  where  there  is  no  chromatic  aberra- 
tion, the  opening  might  be  made  still  smaller,  thus  reducing  the 
brightness  of  the  sky  background. 

The  hypothenuse  of  the  prism  is  silvered,  in  order  that  arti- 
ficial light  entering  through  E  may  fall  upon  the  cell  for  pur- 
poses of  testing  it.  As  soon  as  the  image  of  the  star  is  properly 
adjusted,  the  telescope,  D,  and  the  prism,  are  slid  out  of  the 
way,  so  that  the  beam  of  light  will  fall  upon  the  cell.  FF  is  a 
metal  plate  which  cuts  off  all  light  from  the  cell  when  it  is  not  in 
use.  H  is  a  glass  window,  which  serves  to  protect  the  chamber, 
KK,  as  far  as  possible,  against  the  outside  air.  There  are  screws 
for  adjusting  the  floor  of  the  chamber  so  that  the  cell,  M,  shall 
lie  in  the  optical  axis  of  the  telescope.  J  is  an  opening  into  the 
chamber,  which  can  be  used  for  examining  the  cell  directly. 
The  interior  of  the  cell  is  kept  dry  by  sodium,  which  is  placed 
in  the  receptacle,  L.  A  wire  goes  from  the  cathode  termi- 
nal through  the  amber  plate,  N,  to  the  battery  cells,  while  the 
anode  passes  through  a  similar  plate,  0,  directly  to  the  elec- 
trometer. 

The  electrometer,1  W,  is  suspended  by  a  special  device, 
which  permits  it  to  hang  perpendicularly  no  matter  what  the 
inclination  of  the  telescope  tube.  This  device  is  indicated  in  the 
diagram  by  S,  and  is  itself  supported  on  two  strong  metal  bars, 
QQ,  with  ball  bearings  to  insure  easy  motion.  The  anode  wire 
ends  at  0  in  a  metal  bar,  about  3  mm.  in  diameter,  which  is 
directly  connected  with  the  electrometer  thread,  X.  There  will 
1  Theod.  Wulf,  Physik.  Zeits.,  15,  250. 


PHOTO-ELECTRIC  PHOTOMETRY  163 

be  a  joint  in  this  bar,  as  it  passes  through  the  support,  S,  and 
in  order  to  make  a  perfect  contact,  a  special  connection  is  in- 
troduced, which  is  illustrated  in  the  side  diagram,  V.  The  bar 
ends  in  a  sphere.  In  contact  with  it  is  a  hollow  hemisphere, 
fastened  to  a  hollow  cylinder,  which  rests  upon  the  lower  part 
of  the  bar,  R,  and  is  pressed  against  the  spherical  end  of  the 
upper  part  by  means  of  a  spiral  spring  within,  thus  insuring 
perfect  contact  in  every  position  of  the  telescope.  R  is  protected 
by  a  metal  tube,  U.  The  electrometer  is  a  string  electrometer, 
and  its  thread  is  observed  with  a  micrometer  of  high  magni- 
fying power,  having  an  ocular  scale  in  the  eyepiece,  with  a 
zero-point  in  the  middle  of  the  field.  That  part  of  the  scale  is 
used  which  lies  on  the  right,  or  positive,  side  of  zero,  as  far  as 
division  30  (see  drawing  in  lower  part  of  the  plate).  The  entire 
apparatus  is  protected  at  various  points  by  wires  leading  to  the 
earth,  which  serve  to  carry  away  any  electrostatic  charges 
which  may  collect  upon  it,  due  to  external  causes. 

Briefly  described,  the  operation  of  observing  a  star  is  as  fol- 
lows. After  the  star  has  been  brought  into  the  middle  of  the 
field,  the  telescope,  D,  and  prism,  (7,  are  drawn  out  of  the  way. 
The  connection  between  R  and  the  earth  is  broken,  and  the 
electrometer  thread  begins  to  move. 

Four  times  are  registered  on  the  chronograph.  (1)  Just 
before  breaking  the  ground  connection,  the  position  of  the 
thread  while  at  rest  is  read,  and  the  first  time,  7\,  is  recorded. 
(2)  When  the  thread  passes  over  the  third  or  fifth  division 
farther  on,  the  choice  depending  on  the  rate  of  motion,  the 
second  time,  tlt  is  recorded.  (3)  The  passage  over  the  fifth  or 
tenth  division  beyond  is  recorded  for  the  third  time,  t%.  >  If  the 
zero-point  is  at  division  three,  as  indicated  in  the  diagram,  then 
the  transits  occur  over  the  sixth  and  eleventh  threads,  or  the 
eighth  and  eighteenth  threads,  depending  upon  the  rate  of 
charging.  f  The  thread  is  then  discharged  by  an  earth  connec- 
tion; it  comes  to  rest,  and  the  time,  T&  is  again  noted,  making 
the  fourth  in  the  series.  Before  showing  how  the  reduction  is 
made  from  these  observations  it  will  be  necessary  to  recapitu- 


164          THE  STUDY  OF  VARIABLE  STARS 

late  the  principles  on  which  the  measurement  rests.    The 
authors,  Guthnick  and  Prager,  state:  — 

The  number  of  electrons  liberated,  that  is,  the  observed  photo- 
electric effect  for  alkali  cells,  as  Elster  and  Geitel  have  proved  rigor- 
ously, is  proportional  to  the  intensity  of  the  illumination.  The  meas- 
ured velocities  of  the  electrometer  thread  are  porportional  to  the 
brightnesses  of  the  stars.  This  latter  statement  can  be  considered 
quite  exact  while  the  variations  remain  small. 

The  derivation  of  the  formula  for  applying  this  principle 
is  as  follows  :  — 

Let  H0  be  the  apparent  brightness  of  the  observed  source  of 
light, 

NI  and  2V2  be  the  readings  of  the  zero  point  at  the  times  TI 
and  T2, 

S  be  the  number  of  scale  divisions  passed  over  in  the  inter- 
val of  time  it  —  tit 

A  N  be  the  change  in  the  zero  point  during  this  interval; 

S-AJV 
then  Ho  is  proportional  to  —  -  -  —  . 

12  —  tl 

To  find  A  N  we  have 

2""     *  ==  the  rate  of  change  in  the  zero  reading, 

J-2  —   ll 

A  xr    u 

A  N  =  (t2  - 


j  TJ 

and  Ho  = 


There  follows  the  scheme  of  recording  the  measurements  and 
their  reduction,  only  a  part  of  which  will  be  given. 

1914,  May  22,  comparison  of  7  Bootis  with  B  Urs.  Maj.,  Obs. 
Prager,  S  -  10d,  corr.  to  clock  -  O.m3. 


PHOTO-ELECTRIC  PHOTOMETRY 

a.  The  Measurements 
b.  The  Reduction  of  the  Measurements 


165 


Star 

8  Vrs.  Maj. 

y  Eootis 

8  Urs.  Maj. 

Sid.T. 

17h29m<7 

17h35m.7 

17h45m.2 

Ns-Ni 

-  Op.08 

-  Op.01 

+  9P.18 

T2-Ti 

298.7 

24s.  1 

298.1 

t2-ti 

193.48 

15S.47 

183.98 

(t2-ti):(T2-Ti) 

0.66 

0.64 

0.65 

AN 

-  Op.05 

-  Op.01 

+  Op.12 

log  (s-  AN) 

1.0022 

l.(004 

0.9948 

log  (ti  -  ti) 

1.2896 

1.1895 

1.2783 

log  H0 

9.7126 

9.8109 

9.7165 

[In  this  the  small  value  of  A  N  is  seen  at  once.  From  the  values  of 
log  HQ,  which  are  proportional  to  the  brightnesses,  the  values  of  A  m 
can  be  found  by  Pogson's  rule. 


B 


Aw  = 

=  0.0983 
0.4 


log  A  —  log 
0.4 


loggo-logH'o^  9.8109  -  9.7126 
0.4  0.4 


=  .246. 


If  the  magnitudes  of  y  Bob'tis  and  8  Urs.  Maj.  are  taken  from  the  Pots- 
dam Durchmustening,  the  value  of  A  m  is  found  to  be 
3.34  -  3.52  =  .18.]1 

The  corrections  for  loss  of  light,  "extinction/*  offer  much 
difficulty,  since  they  arise  from  several  different  causes.  Firstly, 
varies  with  the  transparency  of  the  atmosphere;  thirdly,  the 
loss  of  light  is  dependent  upon  the  spectral  type;  secondly,  it 
transparency  of  the  atmosphere  for  the  violet  end  of  the  spec- 
trum is  least  shortly  after  the  end  of  twilight,  and  increases 
during  the  night,  but  not  regularly.  This  is  particularly  true  of 
1  Author's  note. 


166 


THE  STUDY  OF  VARIABLE  STARS 


nights  which  follow  hot  days.  The  entire  matter  requires  much 
further  investigation.  Some  results  of  the  observations  with  the 
photo-micrometer  will  now  be  given. 

Conclusive  proof  is  furnished  that  certain  spectroscopic 
binaries  are  variable  stars  of  very  small  range.  For  instance, 
ft  Cephei,  spectral  type  B  1,  is  a  spectroscopic  binary  not  hith- 
erto suspected  of  variation.  An  exhaustive  discussion  of  a  long 
series  of  comparisons  between  this  star  and  a  Cephei  was  made, 
the  details  of  which  cannot  be  given  here.  The  following  table 
contains  the  co-ordinates  of  the  final  mean  light  curve.  The 
first  column  gives  the  phase,  the  second  the  difference  in  magni- 
tude, (P-a)  Cephei. 


Phase 

0  —  a 

Phase 

|B-« 

d 

m 

d 

m 

0.00 

+  0.142 

0.09 

+  0.188 

0.005 

0.139 

0.10 

0.189 

0.01 

0.140 

0.11 

0.189 

0.02 

0.147 

0.12 

0.187 

0.03 

0.159 

0.13 

0.182 

0.04 

0.171 

0.14 

0.176 

0.05 

0.176 

0.15 

0.170 

0.06 

0.180 

0.16 

0.164 

0.07 

0.183 

0.17 

0.157 

0.08 

0.186 

0.18 

0.150 

0.09 

+  0.188 

0.19 

+  0.142 

Other  stars  investigated  and  found  to  vary  were  a  Can.  Ven. 
and  7  Bootis.  The  same  facts  resulted  from  a  preliminary  study 
of  a  Geminorum  and  o  Persei.  A  long  list  of  stars  which  are 
under  observation  by  Guthnick  and  Prager  follows  in  the  pub- 
lication. It  is  composed  of  many  stars  which  are  either  known 
or  suspected  to  vary. 


PHOTO-ELECTRIC  PHOTOMETRY  167 

The  probable  error  stated  is  not  considered  final,  since  not 
all  of  the  systematic  errors  have  been  eliminated.  The  authors 
give,  however,  as  the  result  from  a  series  of  observations  of  7 
Urs.  Min.  and  f  Dracon.,  the  probable  error  of  a  comparison 
to  be  ±  0.0060  mg. 

It  will  be  seen  that  a  method  has  been  evolved  which  will 
furnish  very  accurate  measurements  of  the  differences  in  mag- 
nitude between  two  stars,  but  that  the  very  sensitiveness  of 
the  apparatus  makes  it  susceptible  to  disturbing  influences, 
and  that  hence  the  sources  of  error  are  large.  Such  an  instru- 
ment can  be  handled  only  by  an  expert.  In  an  earlier  article 
Guthnick  states:  — 

The  apparently  small  interest  which  the  new  method  has  aroused 
among  astronomers  until  quite  recently  can  well  be  explained  by 
the  fact  that  the  physical  and  technical  difficulties  which  must  be 
overcome  are  appalling  to  the  non-physicist,  or  even  cause  him  to 
question  its  success. 

He  himself  acknowledges  his  indebtedness  to  several  physi- 
cists who  aided  him  at  different  points  in  his  work.  The  present 
writer  has  had  to  face  the  above-mentioned  difficulties,  and  has 
depended  upon  her  colleagues  who  are  physicists  to  assist  in 
making  this  explanation  clear,  and  therefore  begs  the  indul- 
gence of  the  reader  if  the  language  is  untechnical  and  the 
presentation  not  complete. 

Before  closing  this  chapter  it  might  be  well  to  bring  to  the 
mind  of  the  reader  in  a  brief  survey  the  chief  events  in  the  his- 
tory of  the  study  of  stellar  magnitude.  Beginning  with  the 
crude  classification  of  Hipparchus,  of  the  lucid  stars  into  six 
divisions  called  magnitudes,  we  find  next  that  Ptolemy  recog- 
nized the  fact  that  all  of  the  stars  of  a  certain  group  were  not 
of  the  same  brightness,  and  assigned  to  those  which  differed 
perceptibly  the  letters  p  and  e,  signifying  greater  than  or  less 
than  the  average.  His  estimations  were  adopted,  and  main- 
tained until  comparatively  recent  times.  Herschel,  in  1780-90, 
felt  that  the  brightness  of  the  stars  ought  to  be  more  carefully 
observed,  and  introduced  symbols,  which  indicated  varying 


168          THE  STUDY  OF  VARIABLE  STARS 

degrees  of  difference,  beginning  with  the  least  perceptible.  In 
the  early  part  of  the  nineteenth  century  photometers  of  vari- 
ous kinds  began  to  be  introduced  into  physical  laboratories, 
and  were  then  applied  to  the  study  of  the  light  of  the  stars, 
but  work  with  them  depended  in  every  case  on  the  magnitudes 
of  stars  already  known.  The  notation  of  Ptolemy  had  given 
way  to  such  notation  as  that  used  by  Argelander,  where  inter- 
mediate grades  were  designated  by  6.5  or  5.6.  Later,  in  working 
on  the  great  Durchmusterung,  where  multitudes  of  stars  passed 
before  the  observer's  eye,  small  differences  became  apparent, 
with  the  result  that  estimations  were  made  directly  to  tenths  of 
magnitudes. 

Photometers  were  improved  from  time  to  time,  until  the 
various  forms  of  polarizing  instruments  reduced  the  error  to  a 
smaller  amount  than  had  been  known  before.  Soon  after  the 
middle  of  the  nineteenth  century  photographic  photometry 
began  to  develop,  bringing  in  its  train  the  study  of  color  and 
spectral  type,  and  their  important  effect  on  the  photographic 
magnitude.  This  ended  in  the  method  of  measuring  the  density 
of  the  star  image  taken  out  of  focus,  which  was  the  last  and  best 
method  of  determining  difference  of  magnitude.  As  a  twentieth- 
century  advance  we  have  instruments  in  which  the  starlight 
falls  upon  electrically  sensitive  surfaces,  such  as  the  selenium 
cell  and  the  photo-electric  cell.  With  the  former  the  secondary 
minimum  of  Algol,  of  only  .06  mg.,  was  discovered,  and  its 
accuracy  could  be  measured  by  the  probable  error  of  a  normal 
of  ±  .006.  The  photo-electric  cell  will  show  the  variation  in 
light  of  a  spectroscopic  binary  when  the  entire  range  is  only  .05 
of  a  magnitude,  and  a  single  comparison  has  a  probable  error 
of  ±  .006. 

Both  of  these  latter  methods  give  results  of  the  very  highest 
degree  of  accuracy,  but  the  manipulation  and  care  of  the  instru- 
ments is  beset  with  difficulties,  so  that  only  an  expert  can 
handle  them.  When  they  have  been  thoroughly  tested,  and 
their  use  has  become  a  little  more  common,  we  can  hope  to 
derive  many  important  results  from  them.  Their  sensitiveness 


PHOTO-ELECTRIC  PHOTOMETRY  169 

makes  them  suitable  for  the  study  of  short  period  variables  of 
rapid  change,  and  for  such  important  and  perplexing  stars  as 
13  Lyrae. 

While  the  eye  cannot  measure  accurately  the  relative  bright- 
ness of  two  stars  of  different  color,  every  photometric  apparatus 
suffers  from  the  same  disability,  and  either  a  correction  must  be 
made  for  the  spectral  type,  or  some  device  introduced  for  mak- 
ing the  colors  of  the  two  objects  alike.  While  the  eye  may  be 
called  our  natural  photometer,  it  is  not  sensitive  enough  to 
very  small  differences  of  brightness,  and  hence  must  be  replaced 
by  more  sensitive  surfaces,  and  itself  be  relegated  to  such  duties 
as  reading  the  deflections  of  a  galvanometer  or  electric  thread; 
so  that  in  place  of  looking  at  a  field  of  stars,  a  beautiful  and 
thrilling  sight  to  the  astronomer,  he  must  perforce  content 
himself  with  reading  a  scale.  As  one  writer 1  puts  it: — 

If  it  should  be  possible  to  develop  a  color  absorbing  screen  which 
should  make  the  resultant  spectral  sensibility  curve  that  of  an  average 
eye,  then  it  should  be  possible  to  tie  down  to  a  purely  physical  instru- 
ment the  characteristics  of  that  wonderful,  but  most  troublesome, 
physiological  one,  —  the  human  eye. 

Both  of  the  methods  described  in  this  chapter  are  too  refined 
for  the  observation  of  long  period  variables,  and  amateur  work- 
ers may  still  plod  along  making  their  observations  by  the 
Argelander  step  method  and  feel  that  their  work,  if  carefully 
done,  will  be  of  scientific  value. 

1  Herbert  E.  Ives,  Ap. «/.,  39,  430. 


CHAPTER  IX 
FORMATION  OF  LIGHT  SCALE 

THE  magnitudes  of  the  comparison  stars  for  a  variable  may 
be  obtained  by  means  of  a  photometer,  basing  the  determina- 
tions upon  a  few  stars  whose  magnitudes  can  be  found  in  some 
one  of  the  recognized  series  such  as  the  Harvard  or  Potsdam 
photometries.  If,  on  the  other  hand,  the  observer  has  no  pho- 
tometer at  his  disposal,  but  must  make  all  of  his  observations 
by  the  Argelander  method,  there  are  two  possibilities  before  him. 
He  may  confine  himself  to  stars  which  can  be  found  on  the 
Hagen  charts,  the  Harvard  photographs,  or  some  other  pub- 
lished map;  or  he  can  study  the  variation  of  light  of  some  new 
variable  for  which  no  list  of  comparison  stars  has  been  pub- 
lished. In  the  latter  case  he  can  determine  the  magnitudes  of 
the  comparison  stars  from  the  observations  by  making  use  of 
the  estimations  of  the  variable,  and  combining  with  them  inter- 
comparisons  among  the  stars  themselves.  Since  this  latter  case 
is  of  very  frequent  occurrence,  it  is  important  to  give  a  full 
description  of  it  here,  with  an  example. 

The  observations  in  the  present  illustration  are  taken  from  a 
collection  made  by  Schonfeld  *  at  the  Mannheim  Observatory 
in  1871,  the  star  being  8  Cephei.  As  given  below  they  are  not 
an  exact  copy  of  the  text.  The  hour  has  been  expressed  as  a 
fraction  of  a  day,  the  letter  v  has  been  used  to  indicate  the 
variable,  and  the  order  of  the  comparisons  has  been  changed  so 
that  the  brightest  star  is  placed  first.  Schonfeld  frequently 
made  use  of  a  quarter  of  a  step,  e.g.,  v  2.5-3  a,  which  is  equiva- 
lent to  v  2.8  a,  but  only  the  final  value  has  been  used  in  the 
example.  Remarks  referring  to  the  brightness  of  the  sky  as 
indicated  by  the  presence  of  the  moon  are  also  omitted.  The 
observations  as  thus  re-arranged  are  as  follows :  — 

1  Dr.  W.  Valentiner,  Veroff.  d.  Grossh.  Sternwarte  zu  Heidelberg,  i,  43. 


FORMATION  OF  LIGHT  SCALE 
TABLE  I 


171 


No. 

Date,  1871 

Ettimation 

£i?A*  itcp 

Mean 

.052XL 

Adopted 

da. 

mg. 

mg. 

I 

May    3.5 

a  I     t>,  r  4     e 

4.0,    4.0 

4.0 

.21 

4.02 

2 

6.5 

t  1.8  v,  v  2.8  a 

7.7,    7.8 

7.8 

.41 

3.82 

3 

7.5 

i  3     v,  v  I     a 

6.5,    6.0 

6.2 

.32 

3.91 

4 

13.5 

a  1.8  v,  v  3.5  e 

3.2,    3.5 

3.4 

.18 

4.05 

5 

14.5 

a  3     v,  v  2      « 

2.0.    2.0 

2.0 

.10 

4.13 

6 

16.5 

f  1     v,  v  0.5  i 

9.9,  10.0 

10.0 

.52 

3.71 

7 

17.5 

i  3     v,  v  1.2  a 

6.5,    6.2 

6.4 

.33 

3.90 

8 

19.5 

a  2.8  v,  v  2.5  e 

2.2,    2.5 

2.4 

.12 

4.11 

9 

21.5 

f  1.2  v,  v        t 

9.7,    9.5 

9.6 

.50 

3.73 

10 

22.5 

i  2.2  v,  v  2.5  a 

7.3,    7.5 

7.4 

.38 

3.85 

11 

23.5 

a  0.2  t>,  t>  5      € 

4.8,    5.0 

4.9 

.25 

3.98 

12 

24.5 

a  2.8  9,  0  2.2  e 

2.2,    2.2 

2.2 

.11 

4.12 

13 

25.5 

a  3.2  0,  v  1.5  e 

1.8,    1.5 

1.6 

.08 

4.15 

14 

28.5 

t  4     t>.  0  1     a 

5.5,    6.0 

5.8 

.30 

3.93 

15 

29.4 

a  2.5  v,  v  2.8  e 

2.5,    2.8 

2.6 

.14 

4.09 

16 

31.5 

a  2     v,  v  2.5  « 

3.0,    2.5 

2.8 

.15 

4.08 

The  first  column  gives  the  number  of  the  observation,  the 
second  the  day,  and  the  third  the  comparison.  The  remaining 
columns  will  be  explained  later.  It  will  be  noticed  first  of  all 
that  four  stars  are  used,  a,  e,  *,  and  f.  On  looking  at  the  list  of 
comparison  stars  which  is  to  be  found  on  p.  264  of  the  same 
volume,  we  find  that  these  are  7  Lacertae,  =  a,  e  Cephei,  t 
Cephei  and  f  Cephei.  In  every  observation  two  of  them  are 
used,  and  by  means  of  the  variable  itself  we  can  find  the 
number  of  steps  between  them.  For  example,  in  the  first 
observation  we  have  a  1  v,  v  4  e,  from  which  we  obtain  a  5  e, 
and  so  on.  Looking  over  the  list  of  observations,  picking  out 


172 


THE  STUDY  OF  VARIABLE  STAES 


the  pairs  of  stars  which  are  used  together,  and  placing  the 
steps  in  order  under  each  pair,  we  find  the  following  combina- 
tions:— 


1.5 
1.2 
1.35 


a  c  i  a 

5.0  4.6 

5.3  4.0 

5.0  4.2 

5.3  4.7 

5.2  5.0 

5.0  4.50 

4.7 

5.3 

4.5 

5.03 

The  next  step  is  to  arrange  these  in  order  so  as  to  form  a  light 
scale  among  the  stars,  which  may  be  done  by  placing  either  the 
faintest  or  the  brightest  star  at  the  bottom  or  zero  end  of  the 
scale.  Both  methods  are  in  use.  Schonfeld  and  Parkhurst 
employ  the  former,  and  Heis  and  Hagen  the  latter.  In  the 
present  case  the  faintest  star  will  be  given  step  zero.  This  is 
obviously  the  star  e.  The  four  when  tabulated  in  order  will 
then  be  as  follows:  — 

TABLE  H 


RHP. 

Star 

L.S. 

Mag. 

Alf 

8494 

«  Cephei 

0.0 

4.23 

.38 
.17 
.06 

8585 

7  Lacertae 

5.0 

3.85 

8694 

i  Cephei 

9.5 

3.68 

8465 

r  Cephei 

10.9 

3.62 

The  fourth  column  in  Table  I  contains  the  result  obtained 
by  making  use  of  the  light  step  of  the  comparison  star  as  found 
in  Table  II  to  find  the  light  step  of  the  variable.  For  example, 


FORMATION  OF  LIGHT  SCALE  173 

in  the  first  observation,  a  1  0,  v  4  e,  a  is  one  step  brighter  than  v, 
but  a  itself  has  a  light  step  of  5.0,  hence  v  must  have  light  step 
4.0  on  the  same  scale.  Also  if  v  is  four  steps  brighter  than  € 
and  e  is  0.0,  then  again  v  has  light  step  4.0.  It  happens  that  the 
comparisons  from  these  two  stars  agree  exactly,  but  such  is 
not  always  the  case,  as  will  be  seen  by  following  down  the  col- 
umn. The  fifth  column  contains  the  mean  of  the  separate 
results  in  the  preceding  column. 

Since  we  have  found  the  brightness  of  the  variable  in  steps 
according  to  the  light  scale,  it  is  possible  to  plot  the  observa- 
tions and  from  them  find  the  character  of  the  curve,  that  is  to 
say,  the  period  and  time  of  maximum  or  minimum.  Instead  of 
this  the  steps  may  be  turned  into  magnitudes,  and  the  results 
plotted  giving  the  light  curve  in  magnitudes.  The  latter  method 
will  be  followed  here. 

The  comparison  stars,  being  all  bright,  are  found  in  the 
Revised  Harvard  Photometry  (RHP.)  Annals,  H.C.O.,  vol.  50, 
from  which  their  numbers  in  Table  II  are  taken,  together  with 
their  magnitudes.  The  difference  in  magnitude  between  suc- 
cessive stars  is  equivalent  to  the  number  of  steps  between  them, 
i.e.,  the  number  of  steps  between  a  and  e  is  5.0,  and  the  differ- 
ence in  magnitude  .38,  hence  to  find  the  value  of  one  step  we 
must  divide  the  difference  in  magnitude  by  the  number  of 
steps.  Proceeding  in  this  manner  we  find  the  following  results 
from  the  different  pairs  of  stars:  — 

a  e=  .076,     i  a  =  .038,  f  i  =  .043;  mean  value  =  .052. 

By  substituting  these  directly  in  the  column  giving  the  light 
step  of  the  variable  we  shall  obtain  its  brightness  expressed  in 
magnitudes.  The  substitution  may  be  represented  by  the 
formula 

M  =  4.23  -  .052  X  L, 

where  L  stands  for  the  light  step  of  the  variable  in  column  5, 
e.g.y  for  the  first  observation, 

M  =  4.23  -  4.0  X  .052  =  4.02. 

The  sixth  column  contains  the  product  .052  X  L.  Its  value, 
when  subtracted  from  the  constant  quantity  4.23  mg.,  will  give 


174          THE  STUDY  OF  VARIABLE  STARS 

the  observed  magnitude  of  the  variable,  which  is  found  in  the 
seventh  column. 

In  plotting  these  observations,  it  is  necessary  first  to  select 
the  scales  according  to  which  the  two  co-ordinates  are  to  be 
drawn.  While  it  is  impossible  to  give  any  specific  advice  on  this 
subject,  in  general  it  may  be  said  that  care  should  be  taken  to 
have  the  resulting  curve  in  a  fair  proportion;  that  is,  it  should 
not  be  too  tall  and  slim,  nor  should  it  be  too  short  and  flat. 
Furthermore,  attention  should  be  paid  to  the  division  of  the 
squares  on  the  paper  used.  A  very  convenient  kind  of  squared 
paper  in  which  every  fifth  line  is  heavier  can  be  purchased  from 
the  various  school  supply  companies.  If  a  large  square  is  taken 
as  a  unit,  then  each  small  one  will  represent  two  tenths.  If  a 
large  square  represents  ten  or  twenty  days,  then  the  small  ones 
will  be  one  fifth  of  this  unit.  A  division  into  tenths  or  fifths  is 
thus  obviously  more  convenient  than  one  into  sixths.  In  the 
curve  given  below,  the  scale  is  as  follows:  for  the  abscissas 
one  large  block  equals  2  days  and  a  small  one  .4  day,  and 
for  the  ordinate  one  large  block  is  .15  mg.  and  a  small  one 
.03  mg. 

If  the  observations  are  sufficiently  well  placed  it  will  then  be 
possible  to  determine  the  time  of  a  maximum  and  the  length  of 
the  period.  In  the  present  case  the  length  of  the  period  can  be 
determined  more  easily  than  the  time  of  maximum,  since  the 
observations  about  May  16  and  21  are  not  well  distributed. 
Still,  since  the  star  is  a  short  period  variable,  the  two  times  of 
brightest  magnitude  could  be  used  for  maxima.  This  point  will 
be  illustrated  more  fully  later  on  in  the  chapter  on  the  mean 
light  curve,  where  this  same  star,  8  Cephei,  is  used  as  an  exam- 
ple. Usually  the  length  of  the  period  is  obtained  by  finding  the 
time  between  successive  maxima,  but  in  this  case  it  may  be 
found  by  taking  the  interval  of  time  between  two  similar  mag- 
nitudes on  corresponding  branches  of  the  curve,  for  example, 
the  star  has  mg.  4.05  on  the  descending  slope  of  the  curve,  on 
the  dates  24.0  and  29.24;  hence  its  period  from  these  two  observ- 
ations is  5.24  days.  A  longer  interval  may  also  be  taken,  as 


9- 


176          THE  STUDY  OF  VARIABLE  STARS 

13.4  to  29.24,  or  15.84  days,  which  is  equivalent  to  three  periods, 
making  one  period  5.28  days  in  length. 

The  usual  method  of  determining  the  time  of  maximum  for  a 
long  period  variable  is,  first,  to  draw  a  smooth  curve  through 
the  observations.  By  this  is  meant  that  there  should  be  as 
many  points  on  one  side  of  the  curve  as  on  the  other  and  that 
the  aggregate  distances  of  the  points  from  it  should  be  equal 
on  both  sides.  To  use  a  familiar  illustration,  it  should  be  like 
a  tug  of  war  where  neither  side  has  the  advantage.  The  pull  on 
one  side  should  balance  the  pull  on  the  other. 

After  the  curve  has  been  finished,  draw  chords  parallel  to  the 
time  axis  and  bisect  them.  Draw  a  line  through  the  points  of 
bisection  and  continue  it  until  it  cuts  the  curve.  The  abscissa 
of  the  point  of  intersection  will  give  the  time  of  maximum,  and 
the  ordinate  the  corresponding  magnitude  or  light  step,  as  the 
case  may  be.  In  drawing  the  chords  care  should  be  taken  to 
have  them  distributed  fairly  well,  with  as  many  as  convenient 
near  the  top  of  the  curve.  The  line  which  passes  through  their 
middle  points  may  itself  be  a  curved  line,  and  there  may  be 
observations  which  give  a  higher  magnitude  than  the  curve 
represents.  Figure  25  is  plotted  from  observations  of  o  Ceti 
made  by  Heis,1  and  published  in  the  collection  of  his  observa- 
tions edited  by  Hagen.  They  extend  from  1848,  Aug.  25,  to 
Jan.  22,  1849,  or  from  Julian  Day  239  6267  to  239  6415,  as  will 
be  seen  in  the  accompanying  Table.  The  variation  is  repre- 
sented in  light  steps,  the  zero  being  taken  for  the  brighter  mag- 
nitudes. Since  the  variable  is  of  long  period,  no  account  is 
taken  of  the  fraction  of  a  day.  Inspection  shows  that  two  points 
at  least  represent  higher  magnitudes  than  the  curve.  Follow- 
ing the  directions  given  above,  for  bisecting  the  chords  and 
drawing  a  line  through  the  points  thus  determined,  we  find  that 
the  time  of  maximum  is  Julian  Day  239  6316  and  the  light 
step  3.3  on  the  given  scale. 
1  Becibaehtungen  ver&nderlicher  Sterne,  by  Eduard  Heis  and  Adalbert  Krueger. 


FORMATION  OF  LIGHT  SCALE 
TABLE  III 


177 


Light  step 

J.D. 

Light  step 

J.D. 

47.5 

239  6267 

6.7 

239  6347 

46.5 

6268 

7.8 

6351 

44.5 

6273 

8.9 

6353 

43.7 

6274 

8.9 

6354 

42.4 

6275 

11.2 

6357 

11.0 

6291 

13.5 

6362 

10.7 

6292 

13.5 

6363 

7.9 

6296 

20.6 

6373 

6.7 

6297 

22.6 

6375 

5.0 

6301 

22.1 

6376 

3.7 

6303 

214 

6380 

4.2 

6307 

25.7 

6382 

4.5 

6321 

26.2 

6383 

3.2 

6323 

28.9 

6384 

4.2 

6326 

29.2 

6385 

5.5 

6333 

28.6 

6386 

5.0 

6334 

32.6 

6393 

5.0 

6339 

45.2 

6415 

Before  this  value  can  be  used  in  further  work,  the  light  steps 
must  be  changed  into  magnitudes.  This  can  be  done  somewhat 
after  the  manner  described  for  8  Cephei,  by  making  use  of  the 
magnitudes  of  the  comparison  stars  as  found  in  the  various 
photometries.  There  is,  however,  another  method,  in  common 
use  at  Harvard,  according  to  which  the  relation  between  star 
magnitude  and  light  step  may  be  obtained.  The  magnitudes 
may  be  plotted  as  abscissas  and  the  light  steps  as  ordinates  of 
points  through  which  a  curve  is  to  be  drawn,  and  this  curve 
may  then  be  used  to  find  either  co-ordinate  when  the  other  is 


F 


FORMATION  OF  LIGHT  SCALE 


179 


given.  The  necessary  material  for  the  example  given  below 
may  be  taken  from  Heis's  observations  of  o  Ceti,  from  the 
beginning  paragraphs,  where  Hagen  has  arranged  in  tabular 
form  the  comparison  stars,  their  light  scale,  and  the  magnitudes 
according  to  several  different  authorities.  Since  this  method  is 
in  such  general  use,  it  seems  advisable  to  take  the  space  here  to 
give  an  illustration.  The  light  steps  are  Hagen's,  and  the  mag- 
nitudes are  taken  from  the  Harvard  Photometry.  They  are  as 
follows :  — 

TABLE  IV 


Star 

Step 

H.P. 

a  Ceti 

0.0 

2.8 

/S  Arietis 

2.3 

2.8 

7  Ceti 

6.2 

3.4 

a  Pise. 

9.6 

3.8 

5  Ceti 

13.2 

4.1 

f  Ceti 

16.8 

4.3 

A*  Ceti 

18.6 

4.3 

P  Ceti 

20.5 

4.6 

XCeti 

23.6 

4.7 

vCeti 

26.7 

4.9 

75  Ceti 

30.6 

5.5 

63  Ceti 

(31.0) 

6.0 

70  Ceti 

33.6 

5.6 

84  Ceti 

34.6 

5.8 

396  B 

37.6 

5.9 

The  accompanying  curve  was  obtained  by  plotting  the  light 
steps  and  magnitudes  as  described  above.  Using  the  maximum 
light  step  3.3,  found  on  p.  176,  we  find  the  corresponding  mag- 
nitude to  be  3.06.  Hence  a  maximum  of  o  Ceti  occurred  on  J.D. 
239  6316,  or  Oct.  15,  1848,  with  a  magnitude  of  3.06. 


180 


THE  STUDY  OF  VARIABLE  STARS 


y 

3.5 

3.0 
2.5 
10 
1.5. 
1.0 

0.5 
0.0 

M£2 

. 

i 

/ 

0* 

) 

7 

/ 

/ 

X 

/ 

5 

X 

/u 

5 

7 

5       ao        3.5       40       4.5       SO       5.5       60 

Figure  26 

MAGNITUDE  CURVE  FOR  o  CETI 

The  example  given  for  forming  the  light  scale  of  the  compari- 
son stars  from  the  observations  of  8  Cephei  involved  only  a  few 
stars,  since  the  variable  was  of  small  range,  and  it  also  happened 
that  the  stars  were  paired  in  the  order  of  their  light  steps.  It 
occurs  sometimes  that  the  different  combinations  overlap,  e.g., 
if  v  were  compared  with  8  and  a,  another  pair  would  be  intro- 
duced, and  the  resulting  step  difference  would  have  to  be  com- 


FORMATION  OF  LIGHT  SCALE  181 

bined  with  the  others.  Sometimes  several  such  groups  occur, 
in  which  case  it  is  necessary  to  alter  somewhat  the  method  of 
procedure,  and  in  computing  the  final  step  for  each  star,  to 
weight  the  individual  means  according  to  the  number  of  ob- 
servations included  in  each  one.  This  may  best  be  illustrated 
by  reference  to  an  example,  and  since  one  has  been  carefully 
worked  out  and  published  in  an  early  number  of  the  Popular 
Astronomy,  by  J.  A.  Parkhurst,  the  present  author  asked 
permission  from  him  to  incorporate  it  in  this  chapter,  and  it  is 
given  practically  entire.  The  example  presents  a  special  advan- 
tage in  that  the  star  studied  is  a  long  period  variable  and  hence 
there  is  a  greater  range  of  magnitude  among  the  comparison 
stars. 

I  will  use  for  an  illustration  my  observations  of  4557  S  Ursae 
Majoris,  from  May  to  December,  1893,  omitting  for  the  sake  of  sim- 
plicity the  comparisons  of  the  comparison  stars  among  themselves. 

Observations  of  4557  S  Ursae  Majoris 

1893  May    11  d2v,  v3f. 

17  d<lvt  t>4/. 

27  d  2  v,  v  3  or  4  /. 

June    19  / 1  v,  v  1  g. 

July       1  h  1 t>,  v  2  k. 

10  h  2  v,  v  k  or  v  1  k. 

Aug.      2    m  2  or  3  v,  n  v  or  n  1  v,  v  3  or  4  o. 

11  n  v  or  n  1  v,  v  1  or  2  o. 

16  n  1  v,  v  1  o. 

26  Moonlight,  v  not  held. 

Sept.     2  n  2  v,  v  o,  v  1  p. 

6  v  about  o,  faint,  seeing  bad. 

26  v  about  m,  difficult,  moonlight. 

Oct.       7  v  m,  v  1  or  2  n,  low. 

9  I  1  or  2  v,  v  2  m,  seeing  good. 

22  g  2  v,  v  3  or  4  I,  v  3  k. 

30  v  3  I.  v  g,  f  2  v,  v  3  k. 

Nov.      2  / 1  v,  v  1  g. 

10  fv,v%g,d*v. 

17  v3f,  dSoriv. 
30  v  5  f,  d  3  v . 

Dec.      3    v  3  /,  d  3  or  4  v . 

First  we  will  form  the  light  scale.  The  observations  of  May  11 
furnish  a  value  for  the  interval  in  steps  between  the  stars  d  and  /. 


182 


THE  STUDY  OF  VARIABLE  STARS 


Since  d  is  2  steps  brighter  than  v  and  v  3  steps  brighter  than  /,  it  fol- 
lows that  d  is  5  steps  brighter  than/.  By  taking  the  mean  of  all  the 
intervals  from  the  observations  in  which  v  is  between  d  and/  in  bright- 
ness, a  good  value  may  be  obtained.  Selecting  similar  combinations 
from  the  observations  of  each  date,  we  can  get  values  for  the  step 
intervals  between  all  the  comparison  stars  used.  Intervals  found  by 
subtraction  are  not  so  reliable,  and  should  only  be  used  when  better 
ones  are  wanting.  The  work  will  stand  as  in  the  following  table,  in 
which  all  the  intervals  between  d  and/  are  ranged  under  the  heading 
df,  and  similarly  for  the  intervals  between  the  other  stars. 


May 


Nov. 


Dec. 


11, 

17. 
27. 
10. 
17. 
30. 
3. 


df 

.5 

.6 

.5.5 

.4 

.6.5 

.8 

.6.5 


7)41.5 
Mean  d  5.9  / 


Aug. 


Sept. 


2, 
11. 
16. 

2. 


Sept.     2 


no 

4 

2 

2 

2 

4)10" 
Mean  n  2.5  o 

np 


June    19  .....................  2 

Oct.     30  .....................  2 

Nov.     2  .....................  2 

10  ....................  .*_ 

4)8_ 
Mean/  2.0  g 


Oct.     22  .....................  5 

30  ....................  .J5_ 

2)8 
Mean  g  4.0  k 


op 


July 


1, 
10, 


Oct.      7. 
Aug.      2, 


hk 

3 

2.5 


Mean  h  2.8  k 

m  n 
..1.5 


mo 
.6 


Sept.     2  ...................  1 


Oct.       9. 


Oct.     30. 


Im 
.3.5 


Oct.     22 5.5 

30 3 

2)8.5 
Mean  g  4.3  / 


kl 
.0.0 


Oct.     30 5 


Oct.     30..  ..5 


dg 


Nov.    10 6 


FORMATION  OF  LIGHT  SCALE 


183 


It  will  be  noticed  that  these  intervals  do  not  exactly  agree  among 
themselves.  For  instance,  we  have  the  intervals/  2  g  and  g  4  k,  from 
which  the  interval  /  6  k  would  result.  But  that  interval,  observed 
directly,  was/  5  k.  Since  the  value/  6  k  depends  on  six  observations, 
while  /  5  k  depends  on  only  one,  by  giving  weights  according  to  the 
number  of  observations  the  mean  value/  5.9  k  would  result.  The  fol- 
lowing method  will  be  convenient  to  make  use  of  all  the  above  inter- 
vals, each  with  its  proper  weight,  in  forming  the  light  scale.  —  Assign 
the  arbitrary  value  o  to  the  faintest  star  used;  in  this  case  p  =  o.  For 
the  next  brighter  star,  o,  we  find  from  the  observation  of  Sept.  2,  the 
interval  o  1  p,  hence  o  =  p  +  1.0  =  1.0.  For  the  next  brighter  star, 
n,  we  have  from  the  above  table, 

n  =  p  +  3.0  =  0.0  +  3.0  =  3.0, 
also  n  =  o  +  2.5  =  1.0  +  2.5  =  3.5. 

These  two  values  for  n  can  be  combined  by  multiplying  each  by 
the  number  of  observations  on  which  it  depends,  and  dividing  the 
sum  of  the  products  by  the  sum  of  the  number  of  observations.  Thus 
n  =  p  _f_  3.0  =  0.0  +  3.0  =  3.0  X  1  =    3.0 
=  o  +  2.5  =  1.0  +  2.5  =  3.5  X  4  =  14.0 

5)17.0 
Mean  n  =  3.4 

By  proceeding  in  this  manner  with  each  brighter  star  successively 
the  scale  values  for  all  will  be  obtained.  The  following  table  shows  all 
the  work  — 


J9=     O 

0=    p+1.0  = 

n=  p_j_3.o  = 


n+1.5  = 
o+6.0  = 


0.0+  1.0=  1.0 
0.0+3.0=3.0X1=    3.0 
1.0+2.5=  3.5X4=  14.0 


3.4+1.5  = 
1.0+6.0  = 


Mean  n=    3.4 

4.9  X  1  =     4.9 
7.0X1=     7.0 


2)11.9 
Meanm=  6.0 


i«m+S.5  =  6.0+3.5=    9.5 

£=    /+0.0  =  9.5+0.0=    9.5 

Ar=fc+2.8=  9.5+2.8=12.3 

f?=   A;+4.0=  9.5+4.0=13.5X2=27.0 

=    /+4.3=  9.5+4.3=13.8X2=27.6 


4)54.6 
Mean?=  13.7 


Light  scale. 
p=  0 
o=  1.0 
n=  3.4 
m—  6.0 
Z=  9.5 
fc=  9.5 
h=  12.3 
g=  13.7 
/=  15.3 
d=  21.0 


184          THE  STUDY  OF  VARIABLE  STARS 

/=  g+  2.0=  13.7+  2.0=  15.7 X  4=  62.8 
=  /+  5.0=  9.5+  5.0=  14.5  X  1=  14.5 
=  &+5.0=  9.5+5.0=  14.5  X  1=  14.5 

6)91.8    • 
Mean/=  15.3 

d=  /+  5.9=  15.3+  5.9=  21.2X  7=  148.4 
=   g+  6.0=  13.7+  6.0=  19.7  X  1=    19.7 

8)168.1 
Mean<f=  21.0 

We  are  now  prepared  to  assign  numerical  values  to  the  brightness 
of  the  variable  at  the  time  of  each  observation.  Here  again  we  must 
take  the  mean  of  slightly  different  values,  for  instance  for  May  11  we 
have  d  2  v>  whence  v  =  19.0,  also  v  3f,  whence  t;  =  18.3;  the  most  prob- 
able value  will  be  the  mean  of  the  two,  or  v  —  18.7.  Proceeding  in  this 
way  with  the  observation  of  each  date  we  have  — 

Date  Observed  Mean 

1893  May     11  19.0,  18.3  18.7 

17  19.0.  19.3  19.2 

27  19.0,  18.8  18.9 

June    19  14.3,  14.7  14.5 

July       1  11.3,  11.5  11.4 

10  10.3,  10.0  10.2 
Aug.      2          3.5,    2.9,  4.5  3.6 

11  2.9,    2.5  2.7 
16          2.4,    2.0                            2.2 
26 

Sept. 

Oct. 


Nov. 


Dec. 

These  results  can  be  represented  to  the  eye  on  squared  paper  by 
laying  off  the  dates  horizontally  and  the  brightness  vertically,  and 
drawing  a  smooth  curve  passing  as  nearly  as  possible  through  the 
points  thus  located.  This  curve  will  show  approximately  the  time  of 
maximum  or  minimum,  and  the  corresponding  brightness  expressed 


2 

1.4,    1.0,1.0 

1.1 

6 

1 

26 

6 

7 

6.0,    4.9 

5.5 

9 

8.0,    8.0 

8.0 

22 

11.7,  13.0,  12.5 

12.4 

30 

12.5,  13.7,  13.3,  12.5 

13.0 

2 

14.3,  14.7 

14.5 

10 

15.3,  15.7,  17.0 

16.0 

17 

18.3,  17.5 

17.9 

30 

20.3,  18.0 

19.2 

3 

18.3,  17.5 

17.9 

FORMATION  OF  LIGHT  SCALE  185 

in  terms  of  the  light  curve.  The  time  of  maximum  or  minimum  can  be 
more  accurately  determined  by  bisecting  the  horizontal  lines  connect- 
ing corresponding  points  on  the  ascending  and  descending  branches  of 
the  curve,  drawing  a  line  through  the  points  thus  located,  and  pro- 
longing it  till  it  intersects  the  light  curve.  This  point  of  intersection 
will  be  the  maximum  or  minimum  as  the  case  may  be. 


CHAPTER  X 

MEAN  LIGHT  CURVE 

AFTER  a  long  series  of  observations  of  a  variable  star  has 
been  made,  and  the  single  light  curves  drawn  from  which  the 
maxima  and  minima  have  been  determined,  the  next  step  in 
order  is  to  combine  them  all  so  as  to  form  the  mean  light  curve. 
While  there  is  no  hard  and  fast  rule  according  to  which  this  is 
done,  the  general  plan  of  procedure  is  the  same,  though  each 
observer  may  introduce  modifications  depending  upon  the 
exigencies  of  his  particular  problem.  The  best  way  to  make  the 
matter  clear  is  to  give  an  extensive  example.  For  this  purpose 
a  group  of  observations  of  8  Cephei,  made  by  Heis,  have  been 
selected  from  those  edited  and  published  by  Hagen,  and  re- 
ferred to  in  Chapter  IX,  which  are  given  below  in  Table  I.  In 
copying  them  some  slight  alterations  in  the  arrangement  have 
been  made  for  the  sake  of  convenience.  The  columns  in  order 
give  the  calendar  date,  the  Greenwich  Mean  Time,  the  Julian 
Day,  with  the  hour  expressed  as  a  fraction  of  a  day,  and  the 
light  step. 

The  first  step  is  to  plot  the  observations  and  draw  the  single 
light  curves.  When  this  has  been  done,  it  will  be  seen  that  in 
some  places  the  observations  are  too  scattered  to  form  a  good 
curve,  but  from  other  sections  where  it  is  well  defined,  the  shape 
can  be  perceived  and  made  use  of  in  the  parts  where  observa- 
tions are  lacking.  We  may  avail  ourselves  of  the  fact  that  the 
light  curve  of  8  Cephei  is  very  well  known,  but  this  knowledge 
cannot  ordinarily  be  expected,  and  in  the  case  of  a  new  short 
period  variable  the  drawing  of  the  single  light  curves  may  be 
attended  with  some  difficulty,  and  several  experimental  curves 
may  need  to  be  drawn  before  a  satisfactory  one  is  obtained. 
Where  the  observations  are  incomplete,  a  dotted  line  is  usually 
drawn.  Since  an  illustration  has  already  been  given  of  the 


MEAN  LIGHT  CURVE 
TABLE  I 


187 


Date,  1848 

Gr.M.T. 

J.D. 

L.8. 

Date,  1348 

Qr.MT, 

J.D. 

L.8. 

d 

h 

239 

d 

h 

239 

June  21 

10.7 

6200.4 

8.3 

Sept.  3 

15.5 

6274.6 

6.8 

22 

10.7 

6201.4 

8.8 

4 

8.1 

6275.3 

7.2 

24 

11.5 

6203.5 

1.7 

5 

9.3 

6276.4 

8.4 

26 

13.3 

6205.6 

7.3 

8 

7.9 

6279.3 

2.0 

27 

10.2 

6206.4 

8.6 

9 

9.2 

6280.4 

8.3 

July  1 

9.9 

6210.4 

7.2 

14 

10.0 

6285.4 

5.9 

2 

13.4 

6211.6 

7.4 

16 

8.4 

6287.4 

7.9 

3 

10.6 

6212.4 

8.2 

17 

9.2 

6288.4 

7.2 

5 

13.2 

6214.6 

2.2 

19 

9.5 

6290.4 

3.3 

6 

10.5 

6215.4 

3.8 

20 

7.3 

6291.3 

7.3 

8 

11.8 

6217.5 

8.1 

21 

7.9 

6292.3 

6.9 

12 

9.8 

6221.4 

6.8 

22 

8.7 

6293.4 

7.9 

13 

9.7 

6222.4 

7.8 

26 

8.6 

6297.4 

7.7 

16 

9.7 

6225.4 

3.9 

29 

7.1 

6300.3 

1.2 

18 

9.3 

6227.4 

8.3 

30 

9.0 

6301.4 

4.2 

22 

10.1 

6231.4 

4.2 

Oct.  2 

7.6 

6303.3 

8.3 

26 

9.8 

6235.4 

2.0 

3 

9.1 

6304.4 

6.7 

27 

9.6 

6236.4 

4.7 

5 

9.6 

6306.4 

3.8 

28 

9.3 

6237.4 

6.5 

6 

7.5 

6307.3 

5.3 

29 

10.0 

6238.4 

7.7 

7 

6.7 

6308.3 

7.8 

30 

9.7 

6239.4 

8.6 

8 

8.8 

6309.4 

8.6 

31 

9.7 

6240.4 

4.7 

15 

6.6 

6316.3 

1.0 

Aug.  1 

10.6 

6241.4 

1.9 

20 

7.2 

6321.3 

1.5 

4 

10.3 

6244.4 

7.1 

22 

7.3 

6323.3 

5.3 

5 

13.8 

6245.6 

4.5 

23 

8.1 

6324.3 

8.3 

10 

14.3 

6250.6 

8.3 

25 

11.1 

6326.5 

3.6 

19 

9.1 

6259.4 

7.2 

26 

6.8 

6327.3 

2.0 

21 

9.0 

6261.4 

8.8 

28 

7.1 

6329.3 

7.7 

22 

9.2 

6262.4 

3.9 

29 

6.0 

6330.2 

8.3 

27 

12.5 

6267.5 

4.2 

Nov.  1 

7.5 

6333.3 

4.2 

28 

8.7 

6268.4 

2.8 

2 

10.2 

6334.4 

7.1 

30 

8.2 

6270.3 

7.7 

188          THE  STUDY  OF  VARIABLE  STARS 

method  of  drawing  a  light  curve,  it  is  unnecessary  to  make  the 
drawing  for  this  star.  However,  the  material  for  drawing  the 
curve  can  be  found  in  Table  I,  and  the  reader  is  advised  to 
make  it  for  himself  and  follow  the  problem  with  it  before 
him. 

The  next  step  is  to  determine  the  approximate  elements,  i.e., 
the  epoch  and  period.  The  epoch  can  be  selected  by  inspection, 
and  is  that  maximum  which  has  the  most  numerous  and  the 
best  distributed  observations.  In  order  to  find  the  period, 
select  three  or  four  of  the  best  maxima,  and  by  combining 
them  in  pairs  obtain  a  fairly  good  approximate  value.  In  the 
present  case,  the  first  maximum  has  observations  both  pre- 
ceding and  following;  hence  it  may  serve  as  the  epoch.  The 
same  is  true  also  of  the  eighth  and  twenty-fourth  maxima, 
but  since  the  first  is  equally  good,  and  is  much  better  situated 
for  the  rest  of  the  work,  it  will  be  chosen  as  the  approximate 
epoch  TQ. 

The  approximate  period  may  be  found  from  the  following 
combinations.  The  interval  in  time  between  the  first  and  eighth 
maxima,  37.9  days,  corresponds  to  seven  periods,  hence  the 
value  of  one  period  will  be  5.41  days.  Combining  the  eighth 
and  twenty-third  maxima  in  the  same  manner,  we  find  an 
interval  of  79.9  days,  which  gives  for  the  period  5.33  days. 
From  the  first  and  twenty-second  maxima  we  derive  a  period 
of  5.37  days.  The  mean  of  these  three  separate  determina- 
tions is  5.37,  which  will  be  accepted  as  the  approximate  pe- 
riod, P0. 

The  next  step  is  to  improve  these  approximate  determina- 
tions by  employing  all  the  other  maxima,  and  there  are  six 
more  which  are  satisfactory  enough  to  be  of  use.  While  it  will 
be  found  that  the  elements  are  not  materially  altered,  the  proc- 
ess will  be  carried  out  in  full,  since  in  the  case  of  other  stars 
this  preliminary  calculation  may  not  give  a  sufficiently  accu- 
rate result.  Before  introducing  the  numerical  values,  the 
equation  will  be  developed  according  to  which  the  corrections 
to  the  approximate  elements  may  be  determined. 


MEAN  LIGHT  CURVE 


189 


Let  TQ  and  P0  be  the  approximate  elements, 
T  and  P  be  the  improved  elements, 
n  stand  for  the  number  of  a  maximum  in  the  entire 

series,  counting  from  TQ,  for  which  n  =  1, 
dT  and  dP  stand  for  the  differences  T  -  T0  and 
P  —  P0,  i.e.,  the  differences  between  the  approxi- 
mate elements  and  the  improved  elements. 
Then  the  two  equations  may  be  formed 

(1)  TQ  +  nP0  =  computed  maximum, 

(2)  (T0  +  dT)  +  n(P0  +  dP)  =  observed  maximum. 
Representing  the  observed  and  computed  maxima  by  the 

symbols  0  and  C,  and  subtracting  the  first  equation  from  the 
second,  we  have  the  resulting  equation 

(3)  dT  +  ndP  =  0-  C. 

Each  observed  maximum  will  give  an  equation  of  this  kind, 
and  the  solution  of  all  of  them  will  give  the  desired  corrections 
to  the  elements. 

In  the  present  case  Table  II  shows  the  necessary  data  for 
forming  the  equations  of  type  (3).  The  first  column  contains 
the  number  of  the  selected  maximum,  the  second  column  the 

TABLE  n 


No. 

O.Max. 

C.  Max 

0-C 

I 

6203.5 

6203.5 

0.0 

3 

6214.6 

6214.2 

+  0.4 

7 

6235.4 

6235.7 

-0.3 

8 

6241.4 

6241.1 

+  0.3 

15 

6279.3 

6278.7 

+  0.6 

19 

6300.3 

6300.2 

+  0.1 

22 

6316.3 

6316.3 

0.0 

23 

6321.3 

6321.6 

-0.3 

24 

6327.3 

6327.0 

+  0.3 

190          THE  STUDY  OF  VARIABLE  STARS 

observed  time  of  maximum,  taken  from  the  observations  in 
Table  I,  column  3.  The  third  column  contains  the  predicted 
maximum,  computed  from  the  elements,  using  equation  (1), 

Computed  Maximum  =  J.D.  239  6203.5  +  n  5.37. 
The  last  column  contains  the  value  "  0  —  C,"  which  is  obtained 
by  subtracting  the  numbers  in  the  third  column  from  the  cor- 
responding ones  in  the  second  column. 

By  substituting  the  numbers  contained  in  the  first  and  last 
columns  of  the  above  table  in  (3)  the  following  equations  are 
formed:  — 

+  1  dT  +    1  dP  =      0.0 

+  1        +3        =+0.4 

+  1        +7        =  -  0.3 

+  1        +8        =  +  0.3 
(4)  +  1        +  15        =  +  0.6 

+  1        +19        =  +  0.1 

+  1        +  22        =0.0 

+  1        +23        =  -  0.3 

+  1        +24        =  +  0.3 

An  inspection  of  the  above  equations  shows  that  they  cannot 
be  solved  by  the  ordinary  algebraic  processes,  since  there  are 
more  equations  than  there  are  unknown  quantities.  Since  this 
is  a  condition  which  usually  occurs  in  astronomical  problems, 
it  is  desirable  to  pause  at  this  point  and  consider  it  somewhat 
in  detail. 

In  any  astronomical  investigation  the  object  is  to  make  a 
series  of  observations  from  which  certain  desired  quantities 
can  be  obtained.  Since  every  observation  involves  some  kind 
of  error,  it  naturally  follows  that  the  greater  the  number  of 
observations  the  more  accurately  the  quantities  sought  can  be 
determined.  The  errors  of  observation  are  of  various  sorts, 
and  depend  upon  the  subject  under  investigation.  When  their 
character  is  understood  theoretically,  their  effects  can  be  com- 
puted and  applied  to  the  observations,  in  which  case  we  say 
that  the  observations  have  been  corrected  for  all  known  errors. 
After  this  has  been  done,  there  are  sometimes  indicated  syste- 


MEAN  LIGHT  CURVE  191 

matic  variations,  the  sources  of  which  may  or  may  not  be  dis- 
covered, but  which  can  usually  be  eliminated  by  some  method 
of  comparison.  Over  and  above  the  theoretical  and  the  syste- 
matic errors,  there  remain  the  accidental  errors,  which  no  kind 
of  foresight  or  study  can  avoid,  and  it  is  with  the  object  of 
eliminating  these  that  observations  are  multiplied,  the  idea 
being  that  in  the  long  run,  small  errors  will  occur  more  fre- 
quently than  large  ones,  and  that  positive  and  negative  errors 
will  occur  with  equal  frequency,  and  hence  neutralize  one 
another.  The  actual  number  of  observations  necessary  to 
make  a  good  determination  will  depend  upon  the  problem. 

Whenever  there  are  more  equations  than  unknown  quanti- 
ties, the  solution  will  be  indefinite,  that  is,  there  will  be  as 
many  solutions  as  there  are  possible  combinations  of  equations 
covering  the  number  of  the  unknowns.  It  therefore  becomes 
necessary  to  devise  some  way  of  getting  around  the  difficulty. 
The  one  adopted  is  called  the  Method  of  Least  Squares.  While 
the  present  book  is  not  the  proper  place  in  which  to  explain  it, 
a  brief  statement  may  be  made  which  will  give  the  ordinary 
reader  some  idea  of  the  principle  on  which  it  is  based. 

Suppose  that  a  set  of  ten  equations,  each  containing  three 
unknown  quantities,  is  given  for  solution,  and  suppose  that  a 
preliminary  result  has  been  obtained  giving  approximate  values 
for  the  three  unknowns.  Suppose,  further,  that  the  values  of 
the  unknowns  are  substituted  in  the  ten  original  equations. 
They  will,  in  general,  not  be  satisfied,  that  is  to  say,  the  second 
terms  will  not  be  zero,  but  a  small  remainder  will  result  from 
each,  which  is  called  a  residual.  Another  approximate  solution 
will  give  another  set  of  residuals,  and  a  third  one  still  another. 
The  question  then  arises,  whether  there  is  any  way  of  deciding 
which  solution  is  the  best.  This  may  be  answered  by  stating 
that  the  residuals  themselves  will  furnish  a  test  as  to  which 
solution  is  the  best;  for,  in  accordance  with  the  Method  of 
Least  Squares,  that  solution  is  the  most  probable  which  makes 
the  sum  of  the  squares  of  the  residuals  a  minimum,  whence  the 
name  "Least  Squares."  There  is  a  regular  method  of  procedure 


192          THE  STUDY  OF  VARIABLE  STAES 

in  the  solution,  which  will  not  be  described  at  this  point,  but 
whose  development  can  be  found  in  special  treatises,  or  in  the 
text-books  of  Chauvenet,  Doolittle,  etc.  It  is  interesting  to 
note  that  in  the  case  of  direct  observations  of  a  single  quantity, 
the  value  which  is  ordinarily  adopted,  viz.,  the  arithmetical 
mean,  is  also  the  most  probable  value  according  to  the  Method 
of  Least  Squares. 

Returning  now  to  the  set  of  equations  (4),  and  considering 
their  solution,  it  is  doubtful  whether  it  is  worth  while  to  carry 
out  the  Method  of  Least  Squares  rigorously,  for  two  reasons: 
first,  the  number  of  equations  is  not  large,  and  secondly,  the 
numerical  terms  are  already  small.  In  such  a  case  the  simplest 
way  would  be  to  group  the  equations  in  two  parts,  four  in  one 
and  five  in  the  other.  There  is  no  rule  according  to  which  this 
should  be  done,  but  since  the  co-efficients  of  dP  increase  suc- 
cessively, it  might  be  better  to  combine  the  alternate  ones. 
By  so  doing  we  obtain  the  two  following  equations:  — 
5  dT  +  69  dP  =  +  0.6, 
4  dT  +  53  dP  =  +  0.5. 
The  solution  of  these  gives 

dT  -  +  0.2454  days, 

dP 0.00909  days, 

and  the  corrected  elements  are  then 

J.D.  239  6203.7454  +  5.36091  E. 

It  is  interesting  to  compare  the  period  obtained  in  this  simple 
manner  from  a  few  observations,  with  that  given  by  Hartwig, 
which  is  5.366386,  being  quite  a  close  approximation.  Except- 
ing for  the  purpose  of  comparison  it  would  be  unnecessary  to 
carry  the  values  of  the  two  unknown  quantities  to  so  many 
decimal  places,  and  for  use  in  the  further  computation  it  will 
be  sufficient  to  adopt  the  values 

Comp.  Max.  =  J.D.  239  6203.745  +  5.361  E. 

The  next  step  toward  the  finding  of  the  mean  light  curve  is 
to  compute  an  ephemeris,  or  a  series  of  maxima  covering  the 
entire  period  of  observation.  In  doing  this  it  is  advisable  to 
use  the  elements  as  given  above,  even  though  in  the  ephemeris 


MEAN  LIGHT  CURVE 


193 


only  one  decimal  place  is  retained.  The  results  obtained  in  this 
way  are  tabulated  below  in  Table  III. 


TABLE  III 


T 

J.D.  Ma*. 

T 

J.D.  Max. 

I 

6203.7 

14 

6273.4 

2 

6209.1 

15 

6278.8 

3 

6214.5 

16 

6284.2 

4 

6219.8 

17 

6289.5 

5 

6225.2 

18 

6294.9 

6 

6230.6 

19 

6300.2 

7 

6235.9 

20 

6305.6 

8 

6241.3 

21 

6311.0 

9 

6246.6 

22 

6316.3 

10 

6252.0 

23 

6321.7 

11 

6257.4 

24 

6327.0 

12 

6262.7 

25 

6332.4 

13 

6268.1 

26 

6337.8 

The  next  step  requires  considerable  care.  Its  purpose  is  to 
locate  each  individual  observation  in  its  light  curve.  A  better 
understanding  of  this  point  may  be  obtained  by  examining  the 
light  curves  which  have  been  depicted  in  Chapter  IX.  A  single 
light  curve  may  be  said  to  extend  from  one  maximum  to  the 
next  one  or  from  one  minimum  to  the  next  one.  A  point  on  the 
curve  can  be  located  in  time  by  giving  the  interval  from  some 
selected  point  to  the  time  of  observation.  This  interval  of  time 
is  called  the  phase.  For  example,  assume  that  in  the  present 
problem  we  shall  count  the  phase  from  the  computed  maximum. 
The  first  maximum  occurs  on  J.D.  6203.7,  the  first  observation 
after  this  is  on  J.D.  6205.6,  therefore  its  phase  is  the  difference 
in  time  between  the  two  times  of  observations,  or  -f  1.9  days. 


194          THE  STUDY  OF  VARIABLE  STARS 

The  observation  preceding  the  same  maximum  is  on  J.D. 
6203.5,  hence  its  phase,  is  —0.2  days.  It  will  be  seen,  then,  that 
we  must  first  find  to  which  particular  maximum  an  observa- 
tion belongs.  In  order  to  decide  this,  we  must  choose  arbitrary 
limits  on  either  side  of  the  maximum  which  will  include  an 
entire  curve.  In  the  case  of  S  Cephei,  the  period  is  approxi- 
mately 5.4  days,  and  the  rise  to  maximum  is  more  rapid  than 
the  descent  to  the  minimum,  therefore  we  shall  choose  the 
limits  from  two  days  before  the  maximum  to  three  and  four- 
tenths  days  afterward,  or  from  —2.0  to  +3.4  days.  Keeping 
this  in  mind,  and  referring  to  the  predicted  maxima  in  Table 
III,  it  is  possible  to  find  the  phase  for  each  observation.  The 
results  thus  obtained  are  arranged  in  Table  IV,  the  first  column 
of  which  gives  the  number  of  the  maximum  in  the  series,  the 
second  the  J.D.  of  the  observation,  the  third  the  phase,  and  the 
fourth  the  light  step.  It  will  be  noticed  that  the  first  two  obser- 
vations occur  more  than  two  days  before  7\,  hence  the  maxi- 
mum immediately  preceding  was  computed  from  the  elements 
and  called  T0,  having  J.D.  6198.4,  but  this  must  not  be  con- 
fused with  the  approximate  epoch  which  was  also  called  T0. 

The  next  step  is  to  rearrange  the  observations  according  to 
the  order  of  the  phase,  selecting  them  for  this  purpose  from  the 
different  light  curves,  wherever  they  happen  to  occur.  They 
are  contained  thus  in  Table  V,  together  with  their  accompany- 
ing light  steps.  The  phases  should  begin  with  the  one  farthest 
preceding  the  maximum,  or  —2.0,  but  with  this  star  it  happens 
that  there  is  no  observation  corresponding  to  this  time,  the 
earliest  phase  being  —1.9.  After  they  are  all  arranged  in  this 
final  order,  the  next  step  is  to  divide  them  into  groups  for  the 
purpose  of  taking  the  averages  for  the  final  mean  light  curve. 
At  this  point  the  observer  must  depend  solely  upon  his  judg- 
ment, and  perhaps  will  have  to  experiment  several  times  before 
obtaining  a  satisfactory  result.  The  points  obtained  from  the 
grouping  must  be  close  enough  together  to  show  the  form  of 
the  curve,  and  yet  not  too  close.  Where  the  curve  is  known 
to  change  its  curvature  rapidly,  the  points  should  be  closer 


MEAN  LIGHT  CURVE 
TABLE  IV 


195 


r 

J.D. 

PAtwe 

L.S. 

T 

J.D. 

Phase 

L.S. 

To 

6200.4 
6201.4 

+  2.0 
+  3.0 

8.3 
8.8 

T« 

6274.6 
6275.3 
6276  4 

+  1.2 
+  1.9 

+     0    /» 

6.8 
7.2 

Q    A 

f»OAQ    K 

—  ft  9. 

1  7 

1 

6205.6 
6206.4 

+  1.9 

+  2.7 

7.3 
8.6 

T15 

6279.3 
6280.4 

+  0.5 
+  1.6 

2.0 
8.3 

T2 

6210.4 
6211.4 
R9.19  4, 

+  1.3 
+  2.3 
4_  q  q 

7.2 
7.4 

Q  £ 

Tlfl 

6285.4 
6287.4 

+  1.2 
+  3.2 

5.9 
7.9 

fiOQQ   A. 

11 

ry  a 

T3 

6214.6 
6215.4 
6217.5 

+  0.1 
+  0.9 
+  3.0 

2.2 
3.8 
8.1 

17 

6290.4 
6291.3 
6292.3 

+  0.9 

+  1.8 
+  2.8 

3.3 
7.3 
6.9 

T4 

6221.4 
6222.4 

+  1.6 

+  2.6 

6.8 
7.8 

T« 

6293.4 
6297.4 

-  1.5 

+  2.5 

7.9 

7.7 

T5 

6225.4 
6227.4 

+  0.2 

+  2.2 

3.9 

8.3 

T18 

6300.3 
6301.4 
fiSOS  S 

+  0.1 
+  1.2 

_L     Q    -1 

1.2 
4.2 

Q  q 

fiOQI    A. 

+  0  8 

A.  a 

fiS04  4 

1  o 

f»  ry 

T7 

6235.4 
6236.4 
6237.4 

COQQ   A. 

-0.5 
+  0.5 
+  1.5 

_i_  a  K 

2.0 
4.7 
6.5 

7  7 

20 

6306.4 
6307.3 
6308.3 

+  0.8 
+  1-7 

+  2.7 

3.8 
5.3 

7.8 

6309  4 

1  fi 

Q   ft 

fiOQQ  4 

—  1  Q 

8  6 

21 

8 

6240.4 
6241  4 

-0.9 
+  01 

4.7 
1  9 

T22 

6316.3 

0.0 

1.0 

6244.4 

+  3.1 

7.1 

T23 

6321.3 

fiQOQ  Q 

-  0.4 

+  1  R 

1.5 

K  Q 

T8 

6245.6 

-  1.0 

4.5 

6324.3 

+  2.6 

8.3 

T10 

6250.6 

-1.4 

8.3 

T24 

6326.5 
6327  3 

-  0.5 
+  ft  3 

3.6 
9  n 

Tu 

6259.4 

+  2.0 

7.2 

6329.3 
6330  2 

+  2.3 

_i_  q  q 

7.7 

0  Q 

(jofji  4, 

_   1    q 

Q.  Q 

12 

6262.4 

-0.3 

3.9 

T25 

6333.3 
fiSS4  4 

+  0.9 
+  9  n 

4.2 
71 

fJ9fi7  £ 

4,  a 

13 

6268.4 
6270.3 

+  0.3 

+  2.2 

2.8 

7.7 

196 


THE  STUDY  OF  VARIABLE  STARS 


together.    Experience  alone  will  indicate  the  best  manner  of 
proceeding. 

TABLE  V 


Phase 

L.S.  • 

Phase 

L.S. 

Phase 

L.8. 

-  1.9 
1.6 
1.5 
1.4 

8.6 
8.6 
7.9 
8.3 

+  0.8 
0.8 
0.9 
0.9 
0.9 

4.2 
3.8 
4.2 
3.3 
3.8 

+  2.2 
2.2 
2.3 
2.3 
2.5 
2.5 
2.6 
2.6 

7.7 
8.3 

7.7 
7.4 

7.7 
7.7 
7.8 
8.3 

-  1.3 
1.2 
1.1 
1.0 
0.9 

8.8 
6.7 
7.2 
4.5 
4.7 

+  1.2 
1.2 
1.2 
1.3 
1.5 
1.6 
1.6 
1.6 

5.9 
6.8 
4.2 
7.2 
6.5 
8.3 
5.3 
6.8 

+  2.7 
2.7 
2.8 
3.0 
3.0 
3.0 

7.8 
8.6 
6.9 
8.8 
8.4 
8.1 

-  0.6 
0.5 
0.5 
0.4 
0.3 

4.2 
3.6 
2.0 
1.5 
3.9 

+  1.7 
1.8 
1.9 
1.9 
2.0 
2.0 
2.0 

5.3 
7.3 
7.3 

7.2 
7.2 
8.3 
7.1 

+  3.1 
3.1 
3.2 
3.2 
3.3 

8.3 
7.1 
7.9 
8.3 
8.2 

-0.2 
0.0 
+  0.1 
0.1 
0.1 

1.7 
1.0 
1.2 

2.2 
1.9 

+  0.2 
0.3 
0.3 
0.5 
0.5 

3.9 
2.8 
2.0 
4.7 
2.0 

The  means  of  the  values  in  the  different  groups  should  now 
be  taken,  the  phase  to  hundredths  of  a  day,  and  the  light  step 
to  hundredths  of  grades.  These  means  are  contained  in  Table 
VI,  page  198,  which  also  includes  other  data  to  be  described 
later.  The  first  column  gives  the  mean  phase,  the  second  the 
mean  light  step,  and  the  third  the  number  of  observations 
included.  The  next  step  is  to  plot  the  observations  and  draw  a 
curve  through  them,  in  doing  which  they  may  be  given  equal 
weight,  or  they  may  be  weighted  according  to  the  number  of 
observations  in  each  group.  If  each  group  is  of  weight  unity, 


MEAN  LIGHT  CURVE  197 

the  process  is  very  simple,  and  the  curve  is  drawn  as  smoothly 
as  possible.  After  this  has  been  done,  the  co-ordinates  of  the 
mean  light  curve  may  be  obtained  by  reading  from  the  one 
just  drawn,  the  light  step  corresponding  to  certain  regular  in- 
tervals, e.g.,  in  the  case  of  8  Cephei,  every  .3  or  .5  of  a  day. 
These  co-ordinates  are  contained  in  the  second  part  of  Table 
VI,  and  comprise  what  is  technically  known  as  the  mean  light 
curve  of  8  Cephei.  That  is  to  say,  when  the  mean  light  curve  of 
a  star  is  spoken  of,  we  may  refer  to  the  co-ordinates  of  the 
curve,  or  to  the  curve  itself.  It  is  frequently  not  convenient  to 
publish  the  curve,  but  if  the  co-ordinates  are  given  they  may 
at  any  time  be  represented  to  the  eye  by  a  drawing. 

Returning  to  the  second  method  of  drawing  the  curve  from 
the  means  in  the  first  part  of  Table  VI,  we  introduce  weights 
which  are  equal  to  the  number  of  observations  in  each  group, 
and  attach  to  each  point  plotted  the  number  representing  its 
weight.  The  points  are  then  connected  by  straight  lines,  and 
each  section  is  divided  into  two  segments,  which  are  inversely 
proportional  to  the  weights  of  the  points.  For  example,  if  the 
first  point  is  the  mean  of  four  observations  and  the  second  the 
mean  of  five,  then  the  line  is  divided  by  a  point  which  is  four 
ninths  of  the  way  from  the  second  to  the  first,  so  that  the  two 
segments  are  in  the  ratio  of  5:4,  the  larger  segment  being 
nearer  the  smaller  weight.  The  points  thus  located  are  then 
connected  by  a  smooth  curve,  which  is  the  mean  light  curve  of 
the  star.  In  the  present  case  the  number  of  observations  in  each 
group  is  so  small  that  it  is  hardly  worth  while  to  use  weights, 
and,  in  fact,  so  far  as  the  author  is  aware,  they  are  generally 
dispensed  with  in  treating  of  discussions  based  upon  visual 
observations  by  the  Argelander  method,  just  as  the  strict 
Method  of  Least  Squares  is  not  generally  introduced,  because 
the  observations  are  not  exact  enough  to  make  it  worth  while 
to  be  so  rigorous. 

The  present  example  may  be  criticized,  and  rightly  so,  be- 
cause light  steps  are  used  throughout  instead  of  magnitudes, 
and  in  any  real  investigation  the  magnitudes  should  be  intro- 


198 


THE  STUDY  OF  VARIABLE  STARS 


duced  at  a  much  earlier  stage  of  the  work,  before  the  single  light 
curves  are  plotted.  However,  the  method  of  procedure  is 
exactly  the  same,  and  since  the  light  step  is  the  value  which 
was  published,  it  was  made  use  of  in  the  example.  Table  VI 
and  the  light  curve  follow. 

TABLE  VI 


M.Ph. 

M.L.S. 

No.  ob». 

Ph. 

L.S. 

da 

da 

-1.60 

8.35 

4 

-1.5 

8.04 

-1.10 

6.38 

5 

1.2 

6.82 

-0.46 

3.04 

5 

0.9 

5.50 

+  0.02 

1.60 

5 

0.6 

3.95 

+  0.36 

3.08 

5 

0.3 

2.39 

+  0.86 

3.86 

5 

0.0 

1.61 

+  1.40 

6.38 

8 

+0.3 

2.81 

+1.90 

7.10 

7 

0.6 

3.48 

+  2.40 

7.82 

8 

0.9 

4.00 

+  2.87 

8.10 

6 

1.2 

5.46 

+3.18 

7.96 

5 

1.5 

6.63 

1.8 

7.09 

2.1 

7.43 

2.4 

7.75 

2.7 

8.02 

3.0 

8.09 

3.3 

7.82    - 

In  recapitulation  the  method  described  at  such  length  may 
be  summarized  as  follows,  with  the  understanding  that  it  may 
be  modified  at  any  point  by  the  observer. 

(1)  Tabulate  the  observations  in  chronological  order,  giving 
the  time  in  Julian  Days. 

(2)  Plot  the  observations  and  draw  the  single  light  curves. 

(3)  Determine  the  times  of  the  observed  maxima  so  far  as 
possible,  and  compute  a  set  of  approximate  elements. 

(4)  Correct  the  approximate  elements  by  using  as  many 
maxima  as  possible,  deriving  the  adopted  elements. 

(5)  Compute  an  ephemeris  of  maxima. 

(6)  Determine  and  tabulate  the  phase  of  each  observation. 

(7)  Rearrange  the  observations  in  order  of  phase. 


MEAN  LIGHT  CURVE 


199 


ML 
1.0 

Z.O 
3.0 
4-0 
5.0 
6-0 
7.0 
8.0 

Da 

.5. 

z 

A 

/ 

\ 

2 

v, 

j 

^ 

s^ 

/ 

\ 

b 

/ 

\ 

i 

/ 

\ 

y 

\ 

2 

5 

/ 

X 

/ 

X 

/ 

>> 

n 

/ 

d 

x>- 

y«    -1.5     -1.0     -0.5       0.0     +0.5     +1.0     +I.S     +Z.Q    +Z.5    +3.0      +3-5 

Figure  27 

MEAN  LIGHT  CURVE  OF  S  CEPHEI 

(8)  Divide  the  observations  into  groups  and  take  the  means. 

(9)  Plot  the  means  and  draw  the  mean  light  curve. 

(10)  From  the  curve  read  off  the  co-ordinates  of  the  mean  light 
curve  for  regular  intervals  of  time. 

When  once  the  mean  light  curve  of  a  variable  star  has 
been  determined  it  can  be  used  for  a  number  of  different 
purposes.  In  fact,  it  is  quite  necessary  when  determining  the 
minima  of  certain  variables  of  the  Algol  type,  as  the  follow- 
ing will  show. 

It  will  be  remembered  that  this  type  is  characterized  by  a 
rapid  descent  to  minimum,  followed  by  an  equally  rapid  rise 
to  maximum,  while  the  duration  of  the  minimum  may  be  quite 
short,  or  it  may  last  for  an  hour  or  more.  Sometimes  the  two 
branches  of  the  curve  are  not  symmetrical,  in  which  case  it  is 
not  possible  to  use  the  ordinary  method  of  bisecting  the  chord 
and  prolonging  the  line  passed  through  the  points  of  bisection 


200 


THE  STUDY  OF  VARIABLE  STARS 


until  it  cuts  the  curve.  We  may  then  make  use  of  the  mean 
light  curve  to  correct  for  the  asymmetry  of  its  two  branches. 
U  Cephei  affords  the  best  example  of  this  method.  The  data 
for  illustration  are  taken  from  an  article  published  by  Yendell 
in  Popular  Astronomy,  14,  600  et  seq. 

TABLE  VII 


T—  t 

Before  mm. 

After  min. 

h  m 

M 

M" 

500 

7.16 

7.18 

440 

7.22 

7.26 

4  20 

7.27 

7.34 

400 

7.34 

7.42 

340 

7.43 

7.49 

320 

7.53 

7.58 

300 

7.64 

7.67 

240 

7.77 

7.75 

220 

7.95 

7.86 

200 

8.18 

8.03 

140 

8.48 

8.29 

120 

8.86 

8.68 

100 

9.05 

9.02 

040 

9.10 

9.06 

020 

9.10 

9.08 

000 

9.09 

The  tabulated  values  give  the  magnitudes  of  the  mean  light 
curve  for  every  twenty  minutes  before  and  after  minimum. 
The  lack  of  symmetry  can  readily  be  seen  by  inspecting  the 
numbers  in  the  table,  or  by  examining  the  curve  in  Chapter  I. 
Yendell  gives  four  observations  made  on  September  11,  1902, 
two  as  the  star  was  diminishing  in  brightness,  and  two  as  it  was 
increasing.  The  times  of  observation  and  the  magnitudes  are 
given  below. 

8h  35m         7.92  mg.       12h     8m         8.23  mg. 

9      0  8.34  12    51  7.68 

By  making  use  of  the  above  table  we  can  find  the  time  of 
minimum  by  a  simple  proportion.  At  the  first  observation  the 
star  had  a  magnitude  of  7.92,  and  was  growing  fainter.  Look- 
ing at  the  tabulated  values,  we  see  that  this  must  have  occurred 


MEAN  LIGHT  CURVE  201 

between  2h  40m  and  2h  20m  before  the  minimum,  as  the 
magnitudes  for  these  two  times  are  7.77  and  7.95.  In  order  to 
find  the  phase  corresponding  to  7.92,  we  must  proportion  be- 
tween the  two  values  as  follows:  the  difference  in  magnitude 
between  7.77  and  7.95  is  .18,  and  the  difference  in  phase  is 
20m,  the  difference  in  magnitude  between  7.77  and  7.92  is  .15, 
while  the  interval  in  time,  x,  is  to  be  determined.  Hence  we 
have 

.18  :  .15  ::  20  :  x 

x  —  16.7m,  and  this  value  must  be  subtracted  from  the  phase 
of  7.77,  which  is  —  2h  40m,  giving  for  the  phase  of  7.92  mg. 
—  2h  23.3m.  That  is  to  say,  the  observation  made  at  8h  35m 
occurred  2h  23.3m  before  the  minimum,  hence  the  resulting 
time  of  minimum  will  be  lOh  58.3m. 

In  the  same  manner  we  find  the  proportions  for  the  other 
three  observations  to  be 

.30  mg.  :  .16  mg.  ::  20  m  :  xm 
.26          :  .06          ::  20       :  x 
.08          :  .07          ::  20       :  x 

which  give  for  the  values  of  x  respectively  10.7m,  4.6m,  and 
17.5m.  The  resulting  times  of  minimum  will  be  lOh  49.3m, 
lOh  23.4m,  and  9h  53.5m.  The  mean  of  all  four  will  be  lOh 
31.1m. 

Chandler,  in  an  article  on  this  star  in  Astronomical  Journal, 
9,  53,  expresses  the  same  fact  in  a  different  way,  giving  a  correc- 
tion which  is  to  be  applied  to  the  point  of  bisection  of  a  chord 
made  in  the  usual  manner.  For  example,  he  finds  that  the  star 
has  a  magnitude  of  8.3,  diminishing,  at  phase  —  Ih  50m,  and 
increasing,  at  phase  +2h  05m,  the  mean  of  which  is  +7.5m. 
This  would  be  applied  as  a  correction  as  follows.  If  the  times 
when  U  Cephei  had  the  magnitude  8.3,  both  diminishing  and 
increasing,  were  observed,  and  the  mean  were  taken,  then  this 
mean  would  differ  from  the  true  time  of  minimum  by  +7.5m, 
and  hence  a  correction  of  —7.5m  would  have  to  be  applied  to 
the  mean  in  order  to  get  the  correct  time.  For  other  magnitudes 
he  finds  the  following  table  of  corrections. 


202          THE  STUDY  OF  VARIABLE  STARS 

mg.       min. 

8.0  0.0 

8.3        -7.5 

8.6        -5.7 

8.9        +0.3 

A  very  interesting  modification  of  the  method  of  this  chapter 
is  used  at  Harvard  in  determining  the  mean  light  curves  of  a 
series  of  long  period  variables  as  published  in  Annals,  H.C.O., 
37.  The  observations  which  appear  in  these  volumes  were 
made  by  different  observers  at  different  places,  hence  the 
aggregate  number  is  very  large,  and  they  are  scattered  quite 
thickly  along  most  parts  of  the  curve.  Since  it  was  difficult  to 
handle  so  many  observations,  the  proceeding  was  as  follows. 
The  observations  were  all  plotted  as  in  the  case  of  $  Cephei, 
and  a  smooth  curve  was  drawn  through  them,  making  the  suc- 
cessive branches  as  much  alike  as  possible.  This  is  not  so  simple 
as  with  short  period  variables,  since  the  long  period  variables 
do  not  always  reach  the  same  magnitude  at  maximum  or  mini- 
mum, and  the  periods  are  not  of  uniform  length.  After  the 
curves  had  been  drawn  satisfactorily,  the  magnitudes  were 
read  from  them  for  every  twenty  days,  counting  the  multiples 
of  twenty,  e.g.,  J.D.  1600,  1620,  1640,  etc.,  and  tabulated. 
These  magnitudes  were  used  afterward  in  place  of  the  original 
observations.  The  process  of  finding  the  time  of  maximum  and 
minimum  was  also  different,  use  being  made  of  tabulated  values 
instead  of  bisecting  the  chords,  as  is  the  usual  custom.  A  second 
table  was  formed,  containing  the  dates  for  every  half  magni- 
tude on  the  ascending  and  descending  branches  of  the  curves, 
and  the  middle  points  taken.  These  dates  were  then  plotted 
with  the  corresponding  magnitudes  as  ordinates,  and  lines 
drawn  through  the  points  and  extended  until  the  maximum 
magnitude  was  obtained.  As  will  be  seen,  this  is  practically  the 
same  method  as  bisecting  the  chords. 

Other  differences  in  method  entered  when  the  phase  of  an 
observation  was  found,  the  chief  one  being  due  to  the  fact  that 
with  these  variables,  the  periods  are  not  of  uniform  length,  nor 


MEAN  LIGHT  CURVE  203 

is  the  time  from  maximum  to  minimum  the  same,  so  that  the 
observations  cannot  be  arranged  in  regular  order,  from  the 
maximum  throughout  an  entire  period.  Hence  they  were 
counted  both  ways  from  the  maximum  and  both  ways  from  the 
minimum,  care  being  taken  that  the  entire  curve  was  covered. 
The  two  sections  were  then  put  together  according  to  the  mean 
value  of  M  —  m.  As  an  illustration  of  this  point  the  values  for 
T  Cassiopeiae  may  be  cited.  The  mean  period  of  this  star  is 
445  days,  and  the  interval  M  —  m  is  261  days,  leaving  184  days 
for  the  other  part  of  the  period.  The  maximum  part  of  the 
curve  may  be  considered  to  extend  from  —130  days  to  +92 
days,  and  the  minimum  from  —92  days  to  +130  days.  In 
order  that  the  two  branches  shall  overlap  somewhat,  and  since 
the  observations  to  be  used  are  tabulated  for  every  20  days, 
the  phases  should  extend  from  140  to  100  days  on  either  side 
of  the  initial  point. 

While  it  would  be  very  interesting  to  carry  through  com- 
pletely the  investigation  of  one  of  these  long  period  variables, 
space  does  not  permit  it;  furthermore  the  problem  is  worked 
out  in  detail  in  the  volume  of  the  Annals  just  quoted,  and  any 
investigator  who  is  engaged  in  such  work  can  easily  procure 
the  volume  and  follow  it  for  himself. 

Many  other  modifications  of  the  process  are  doubtless  in  use 
by  different  observers,  which  may  be  found  and  studied  in 
various  publications,  but  the  steps  to  be  followed  are  in  general 
those  indicated  in  this  chapter.  , 


CHAPTER  XI 

PREDICTION  OF  MAXIMA  AND  MINIMA  FROM  THE 
ELEMENTS 

IN  computing  the  maximum  or  minimum  of  a  variable  star 
from  its  elements  the  observer  has  two  objects  in  view;  the  first 
is  to  compare  the  observed  with  the  computed  dates,  in  order 
to  confirm  the  accuracy  of  the  elements,  and  the  second  is  to 
predict  the  dates  in  order  to  prepare  lists  for  observation,  such 
as  have  already  been  referred  to  in  describing  the  ephemerides 
published  yearly  by  Hartwig.  Predictions  are  also  published 
monthly  in  the  Popular  Astronomy  and  in  other  places,  but  the 
regular  observer  should  be  able  to  make  these  computations  for 
himself;  hence  examples  of  the  different  methods  will  be  given. 
However,  before  taking  up  a  discussion  of  the  different  formu- 
las employed  to  express  the  variation  of  a  star,  it  is  desirable 
to  consider  at  some  length  the  subject  of  Julian  Day,  already 
mentioned  more  than  once,  and  to  give  rules  for  its  use. 

We  are  accustomed  to  associate  the  term  Julian  Calendar 
with  the  reform  introduced  by  Julius  Caesar,  according  to  which 
the  year  is  made  to  consist  of  365.25  days,  and  in  the  course 
of  four  years  the  fractional  part  amounts  to  one  whole  day. 
Hence  one  day  is  added  to  every  fourth  year,  making  it  the 
leap  year.  Unfortunately  for  our  convenience,  the  number  just 
given  is  not  exact,  for  the  tropical  year  contains  365.2422  days. 
Consequently  if  a  day  is  added  every  four  years,  in  the  course 
of  time  the  error  will  reach  several  days,  and  the  return  of  the 
seasonal  festivals,  which  have  been  considered  very  important 
dates,  will  come  out  of  time.  The  divergence  from  the  truth 
can  readily  be  perceived  by  a  simple  calculation. 

A  quadrennium  is  the  name  given  to  a  period  of  four 
years  consisting  of  three  common  years  and  one  leap  year, 
or  1461  days.  Four  years  of  the  correct  length  will  consist  of 


PREDICTION  OF  MAXIMA  AND  MINIMA    205 

4  X  365.2422  days,  or  1460.9688  days,  which  differs  from  1461 
by  .0312.  Hence  adding  a  day  every  four  years  makes  the 
quadrennium  too  long  by  this  amount,  or  each  year  is  .0078 
day  too  great.  To  find  out  how  soon  this  will  amount  to  an 
extra  day,  divide  1  by  .0078  and  the  result  is  128,  therefore  the 
128th  year,  instead  of  being  a  leap  year,  should  be  a  common 
year,  and  so  on.  This,  however,  would  be  a  rather  inconvenient 
and  irregular  way  of  making  the  correction,  and  a  simpler  one 
has  been  adopted.  The  extra  time  amounts  to  .0312  day  every 
four  years,  or  to  3.12  days  in  400  years,  hence  if  three  leap 
years  are  omitted  every  400  years,  the  agreement  will  be  very 
nearly  exact.  When  the  calendar  was  reformed,  under  Pope 
Gregory,  in  the  sixteenth  century,  it  was  decided  that  this 
could  be  accomplished  by  calling  the  centuries  leap  years  only 
when  they  were  divisible  by  400  and  not  otherwise.  Therefore 
1700,  1800,  and  1900  were  not  leap  years,  but  2000  will  be  one. 
The  agreement  is  not  even  thus  perfect,  for  .12  day  will  have 
been  added  every  400  years,  which  will  amount  to  a  day  in 
3333  years,  but  this  is  such  a  negligible  quantity  that  for  pres- 
ent purposes  the  Gregorian  Calendar  may  be  considered  exact. 
It  was  introduced  into  the  Catholic  countries  in  1582,  at  which 
time  the  difference  amounted  to  ten  days,  and  October  4  was 
followed  by  October  15.  It  was  not  adopted  in  England  and 
her  colonies  until  the  eighteenth  century,  and  September  2, 
1752,  was  followed  by  September  14,  as  the  difference  had 
increased  by  one  day  more.  Hence  in  reality  George  Washing- 
ton was  born  on  February  11,  1732.  The  Julian  Calendar  is 
still  in  use  in  Russia,  and  letters  are  frequently  headed  with 
both  dates,  as  "January  15/28,  1914."  Often  a  date  given 
according  to  the  Julian  Calendar  is  designated  as  O.S.,  or  Old 
Style. 

The  Julian  Period  is  a  certain  length  of  time  arbitrarily 
adopted  to  cover  the  duration  of  historical  records,  and  counted 
according  to  the  Julian  system.  It  includes  a  cycle  of  7980 
Julian  years,  and  began  on  the  noon  of  January  1,  B.C.  4713. 
This  number  is  based  upon  three  subordinate  cycles,  which  are 


206          THE  STUDY  OF  VARIABLE  STARS 

the  Solar  Cycle,  the  Lunar  Cycle,  and  the  Cycle  of  Indictions. 
The  first  one  consisted  of  twenty-eight  Julian  years,  the  second 
of  nineteen,  and  the  third  of  fifteen,  and  the  entire  cycle,  which 
must  include  all  three,  will  be  the  least  common  multiple  of 
them,  which  is  7980.  The  year  B.C.  4713  was  selected  as  the 
origin  of  the  period,  since  it  is  the  year  which  was  number  one 
in  each  of  the  subordinate  cycles.  Starting  from  this  date  the 
days  which  have  elapsed  can  be  computed  simply  and  regu- 
larly according  to  the  Julian  Calendar.  Tables  have  been  con- 
structed for  converting  a  calendar  date  into  Julian  Days  and 
vice  versa.  The  extreme  convenience  of  this  way  of  reckoning 
can  readily  be  understood  when  we  attempt  to  combine  observ- 
ations of  variable  stars  made  over  a  long  interval  of  time, 
where  the  awkwardness  of  using  calendar  dates  becomes  quite 
obvious.  It  was  first  introduced  into  variable  star  work  by 
Pickering1  in  1890.  Nearly  all  series  of  observations  are  pub- 
lished with  the  calendar  date  and  the  Julian  Day,  which  is 
designated  by  the  familiar  abbreviation  of  "  J.D."  The  three 
subordinate  cycles  are  still  used  in  determining  the  dates  of 
festivals,  such  as  Easter,  and  the  American  Ephemeris  gives 
for  every  year  its  cyclical  numbers,  e.g.,  1916,  Lunar  Cycle  or 
Golden  Number,  17;  Solar  Cycle,  21;  Roman  Indiction,  14; 
Julian  Period,  6629.  Table  I  at  the  end  of  this  volume  furnishes 
the  material  for  converting  a  calendar  date  into  Julian  Days, 
the  use  of  which  is  explained  in  the  introduction  to  the  tables. 
We  shall  now  proceed  to  give  illustrations  of  computing  the 
maximum  or  minimum  of  a  variable  star  from  its  elements, 
taking  the  data  from  Hartwig's  Ephemeriden.  On  looking  over 
the  column  in  Table  I,  which  gives  the  elements,  it  will  be 
noted  that  in  general  only  two  terms  are  given,  the  epoch,  from 
which  the  number  of  periods  is  counted,  and  the  length  of  the 
period  given  in  days.  For  example,  in  the  volume  for  1914, 
from  which  all  of  the  examples  will  be  taken,  No.  1,  SS  Cass., 
has  for  its  elements 

2417504  -  139.6  E, 
>  Annals,  H.C.O.,  18,  305.  % 


PREDICTION  OF  MAXIMA  AND  MINIMA    207 

which  means  that  J.D.  2417504  is  the  Julian  Day  of  some  well- 
observed  maximum,  139.6  days  is  the  length  of  the  period, 
and  E  is  the  number  of  periods  which  has  elapsed  since  the 
epoch.  Star  13,  T  Cass.,  has,  in  addition  to  the  ordinary  ele- 
ments, a  sine  term  which  indicates  a  periodic  variation  in  the 
period  itself.  The  same  sort  of  additional  term  appears  in 
several  places.  R  Virginis,  No.  418,  has  two  sine  terms.  Some 
stars  like  No.  151,  S  Tauri,  have  a  term  in  E 2,  which  signifies 
that  the  variation  in  the  length  of  the  period  is  secular  and  not 
periodic,  that  is,  it  goes  on  in  the  same  direction  without 
change.  One  star,  R  Hydrae,  No.  437,  has  both  periodic  and 
secular  terms,  as  does  also  S  Serpentis,  No.  484.  Such  a  com- 
bination is  by  no  means  simple  to  work  out,  but  the  method 
should  be  understood. 

Suppose  it  is  desired  to  compute  the  maximum  of  a  variable 
star  for  1914,  the  first  step  will  be  to  find  the  value  of  E.  If  the 
observer  has  made  a  similar  prediction  for  preceding  years,  it 
will  only  be  necessary  to  add  1  to  the  number  used  for  1913, 
but  if  the  work  is  to  be  done  de  novo  the  value  for  E  must  be 
determined  by  a  method  which  is  more  or  less  approximate, 
because  of  the  presence  of  the  additional  terms  and  their 
numerical  values.  That  is  to  say,  a  preliminary  value  of  E 
must  be  found,  substituted  in  the  formula,  and  the  Julian  Day 
for  the  result  determined  from  the  table;  if  the  date  thus  found 
does  not  fall  in  the  year  1914  the  value  for  E  must  be  changed 
and  the  computation  repeated.  As  a  rule  not  more  than  one 
such  additional  computation  is  required,  especially  when  the 
elements  consist  of  but  one  term.  Several  examples  will  now 
be  given  to  illustrate  the  different  cases. 

METHOD   OF   FINDING  E 

E  will  always  be  a  whole  number,  since  it  represents  the 
recurrence  of  successive  maxima,  each  of  which  is  an  integral 
number  of  periods  from  the  epoch.  The  maximum  which 
occurs  during  1914  will  be  the  last  one  preceding  the  begin- 
ning of  1915.  Therefore  if  we  find  the  interval  of  time  between 


208          THE  STUDY  OF  VARIABLE  STARS 

the  epoch  and  January  0, 1915,  and  divide  by  the  length  of  the 
period,  we  shall  find  the  number  of  maxima  which  have 
occurred  during  that  time.  If  the  period  is  much  less  than  a 
year,  and  two  maxima  take  place  during  1914,  the  one  thus 
found  will  be  the  last  one  for  the  year,  but  the  preceding  one 
may  be  found  by  subtracting  1  from  E.  For  convenience  in 
carrying  out  these  computations,  the  two  Julian  Days  most 
necessary,  those  for  January  0,  1914,  and  January  0,  1915,  will 
be  given  here.  They  are  respectively,  2420133  =  T0,  and 
2420498.  (H)  stands  for  the  predicted  value  taken  from 
Hartwig's  Ephemerlden  for  1914. 

(1)      Maximum  of  No.  1,  SS  Cass.   2417504  +  139.6  E. 


Jan.     0,  1915 
Epoch 
Diff. 
E 

P             139.6 
E                 21 

2420498 
2417504 

Epoch             2417504 
PXE                  2931.6 
2420435.6 
T0                   2420133 

2994)139.6 
21 

Oct.     29.6              302.6 
Oct.    30  (H) 

2931.6 

Since  the  period  is  short,  two  other  maxima  will  also  occur 
during  the  year,  one  at  139.6  days  earlier  than  October  30,  or 
163  from  the  beginning  of  the  year,  i.e.,  June  12,  and  another 
one  139.6  days  still  earlier,  i.e.,  January  23. 

(2)       Maximum  of  No.  2,  TT  Cass.   2418588  +  398  E. 

Jan.     0,  1915         2420498  Epoch  2418588 

Epoch  2418588  PXE  1592 

Diff.  1910)398  2420180 

E  4  T0  2420133 

Feb.     16  47 

Feb.    16  (H) 
P  398 

E  4 

1592 


PREDICTION  OF  MAXIMA  AND  MINIMA    200 

(3)     Maximum  of  No.  976,  SS  Pegasi.    2418865  +  412  E. 

Since  the  period  is  greater  than  365  days  it  may  happen  that 
no  maximum  occurs  during  the  year  1914.  If  this  is  true,  the 
formula  will  give  as  a  result  a  number  less  than  the  J.D.  for 
January  0,  1914. 


Jan.  0, 
Epoch 
Diff. 
E 

P 
E 

1915 

412 
3 
1236 

2420498 
2418865 

Epoch 
PXE 

Jan.  0, 

Nov.  29 
Nov.  29  (H) 

«      ' 

2418865 
1236 
2420101 
2419768 

1633)412 
3  1913, 

1913, 

333 

The  next  maximum  will  occur  412  days  later,  which  will  not 
be  until  1915. 

(4)  Maximum   of  No.    17,  R  Androm.    2402596  +  410.64 

+  30  sin  (12  £7 +168). 

To  obtain  the  preliminary  value  for  E  we  can  use  only  the 
first  term  (1)  for  the  period.  If  the  addition  of  the  second 
term  (2)  throws  the  maximum  into  the  adjacent  year,  the 
value  for  E  must  be  changed. 

Jan.   0,  1915  2420498 

Epoch  2402596 

Diff.  17902)410.64 

E  43 

P          .  410.64 

E         43 

(1)        17657.52 

sin  324° 
Epoch  2422596 

(1)  + 17657.5  (2) 

(2)  -        17.7 

2420235.8 

T0  2420133 

Apr.  12.8  102.8 

Apr.  13     (H) 


210 


THE  STUDY  OF  VARIABLE  STARS 


(5)  Maximum    of  No.  151,   S  Tauri. 
-0.15  E2. 


2400455  +  380.0  E 


Jan.  0, 

1915           2420498 

Epoch 

2400455 

Epoch 

2400455 

(1) 

•f  19760 

Diff. 

20048)380.0 

(2) 

-      405.6 

E 

52 

2419809.4 

r. 

2420133 

P 

380.0 

E 

52 

(1) 

19760 

E» 

2704 

-.15 

(2)   ~ 

405.60 

The  result 

indicates  that  the  value  of  E  is  too  small,  since  it 

places  the  maximum  before  January 

0,  1914. 

Accordingly 

the 

computation 

must  be  repeated,  using  E  =  53. 

P 

380.0 

Epoch 

2400455 

E 

53 

(1) 

H-  20140.0 

(1) 

20140.0 

(2) 

-      421.35 

2420173.65 

To 

2420133 

E« 

2809 

Feb.  9.65 

40.65 

-  .15 

(2) 

-  421.35 

Feb.  10  (H) 

PREDICTION  OF  MAXIMA  AND  MINIMA    211 


(6)     Maximum  of  No.  418,  R  Virginis.    2381934.8  +  145.47  E 
+  20  sin  (1°.8  E  +  216°)  +  4.8  sin  (5°.625  E  +  343°). 


Jan.  0,  1915 

2420498 

Epoch 

2381935 

Diff. 

38563)145.47 

E 

265 

P     145.47 

E        265 

(1)    38549.55 

E 


265 


333° 


(2) 


-  9.00 


5°.625 

E 

265 

1490.625 

343 

1833.625 

4X360° 

1800 

33°.625 

sin  33°.625 

+.55 

(3) 

4.8 

+  2.640 

Epoch 

2381935 

(1) 

(2) 

+  38549.55 
-         9.00 

(3) 

-f          2.64 

2420478.19 

T. 

Dec.  11 

2420133 

345.19 

Dec.  11  (H) 

(7)    Maximum  of  No.  437,  R  Hydrae. 
-  0.36  E2  +  15  sin  (7°.5  E  -  202°). 

Jan.     0,  1915 

Epoch 

Diff. 


2411931  +  425.15  E 


E 

P 
E 

(1) 
E* 

(2) 


425.15 

20 

8503.00 


2420498 

7°.5 

2411931 

20 

8567)425.15 

150.0 

20 

202 

352°.0 

sin  352° 

-    .14 

(3) 

15 

-2.10 

Epoch 

2411931 

(1) 
(2) 
(3) 

+  8503.0 
-      144.0 
-         2.1 

2420288.9 

r0 

2420133 

June  3.9           154.9 

June  4 

(H) 

212          THE  STUDY  OF  VARIABLE  STARS 

The  preceding  examples  include  nearly  all  the  varieties  of 
method  required  for  computing  the  maximum  of  a  long  period 
variable.  In  order  to  obtain  the  time  of  minimum  which  is 
predicted  for  many  of  the  stars  in  Hartwig's  Ephemeriden  it  is 
necessary  to  know  the  interval  of  time  between  the  minimum 
and  the  following  maximum,  or  M  —  m.  This  is  not  given  by 
Hartwig,  but  may  be  found  in  Chandler's  catalogues  for  some 
of  the  stars.  It  is  also  found  in  Annals,  H.C.O.,  vol.  55,  not  as 
a  part  of  the  catalogue,  but  in  Table  VII,  together  with  other 
material.  The  predicted  time  of  minimum  can  be  found  from 
the  time  of  maximum  by  subtracting  the  interval  M  —  m  from 
it  and  finding  the  J.D.  of  the  result.  It  will  be  noted  that  this 
value  is  lacking  for  a  great  many  long  period  variables,  and  the 
necessity  of  supplying  this  lack  points  to  a  very  useful  kind  of 
observation  for  those  who  can  work  with  a  telescope  large 
enough  to  show  stars  of  the  thirteenth  magnitude,  and  can 
follow  these  variables  through  their  minima. 

The  process  of  finding  the  maximum  or  minimum  of  a  short 
period  variable  is  similar  to  that  in  use  for  the  long  period 
variables.  To  illustrate  this  a  minimum  of  Algol  will  be  com- 
puted. The  elements  which  are  found  in  Hartwig  are  taken 
from  Chandler's  catalogue,  and  are  given  in  two  forms;  the 
first  gives  the  calendar  date  for  the  initial  epoch  and  expresses 
the  period  in  hours,  minutes,  and  seconds,  while  in  the  second 
these  quantities  are  given  in  fractions  of  a  day.  The  second 
form  is  the  one  most  convenient  to  use  in  predicting:  — 


m  =  J.D.   2410640.34111  +  2d.8673102  E 

53  sin  < 

13 


7?° 

Od.1021  sin  (0°.024  E  +  226°)  +  Od.0153  sin  (—+  216°) 


In  finding  the  value  for  E  the  last  two  decimal  places  in  the 
period  may  be  omitted,  but  after  that  the  complete  number 
must  be  used,  and  the  multiplication  done  in  full,  and  not 
by  logarithms,  to  avoid  the  possible  inaccuracy  in  the  last 
figures. 


PREDICTION  OF  MAXIMA  AND  MINIMA    213 

Jan.     0,  1915        2420498  E/13  264 

Epoch  2410640  216 

Diff.  9858)2.86731  480 

E  3438  JK50 

120 

P  2.8673102 

E  3438  sin  120°          +  .87 

(1)  9857.8124678  .0153 

(3)  +  .013311  d 

E 

Epoch     2410640.34111 

(1)  +     9857.8124678 

(2)  -  0.079638 
808°.5                                (3)         +  0.013311 

2420498.08725 

sin  308°.5  -  .78  Jan.  0.08725= 

.1021 

(2)  -  .079638  d  Dec.     31,  2h  5m  388.4 

Dec.     31,  2h.l  (H) 

In  changing  the  fraction  of  a  day  to  hours,  minutes,  and  sec- 
onds, use  is  made  of  Table  II. 

It  will  be  noticed  that  the  predicted  times  are  called  "Helio- 
centric minima  for  Greenwich  Mean  Time."  The  observed 
times,  however,  are  geocentric,  and  must  be  corrected  for  the 
difference  between  the  two  before  being  compared  with  the 
computed  times.  Hartwig,  in  the  explanation  of  his  tables, 
page  293,  supplies  a  formula  for  this  purpose  the  derivation 
of  which  will  be  given  here. 

In  the  accompanying  figure,  on  page  214,  let  XEY  be  the 
plane  of  the  ecliptic,  the  axis  of  X  pointing  to  the  vernal 
equinox,  <S,  is  the  position  of  the  sun  in  this  plane,  Ecr  the 
line  of  sight  to  the  star,  and  So;  which  is  parallel  to  Ecr,  the 
line  from  the  sun  to  the  star.  Project  ES  on  Ear,  forming  EH. 
Then  the  instant  of  time  when  the  light  from  the  star  reaches 
H  will  be  the  same  as  when  it  reaches  S  and  will  be  earlier 
than  the  instant  it  reaches  E,  by  the  length  of  time  required 
to  pass  over  the  distance  EH.  The  problem  then  is  to  find 
the  time  required  by  light  to  pass  from  H  to  E. 


214          THE  STUDY  OF  VARIABLE  STARS 


Figure  28 

DIAGRAM  FOR  OBTAINING  THE  REDUCTION  TO  THE  SUN 

Pass  a  plane  through  E<r  perpendicular  to  the  plane  of  the 
ecliptic,  cutting  it  in  the  line  ED.  Let  E  be  the  center  of  a 
sphere  passing  through  S.  Then  in  the  figure  we  have  the 
following  relations:  — 

XES  =  0  =  the  longitude  of  the  sun, 
XED  =  X  =  the  longitude  of  the  star, 
SED  =  \-  O, 
SEH  =  a  =  M  S, 

ES  =  R,  the  radius  vector  of  the  earth, 
EHS  =  90°, 

M  ED  =  ft  =  the  latitude  of  the  star, 
MDS  =  90°. 
In  the  plane  triangle  EHS 

EH  =  R  cos  a. 


PREDICTION  OF  MAXIMA  AND  MINIMA 

In  the  spherical  triangle  MDS 

cos  a  =  cos  @  cos(\—  O). 
Hence  EH  =  R  cos  £  cos(\-  O ) . 

In  this  equation  EH  is  expressed  in  the  same  unit  as  R.  If  we 
wish  to  find  the  time  required  by  light  to  travel  over  this  dis- 
tance, we  must  divide  both  members  of  the  equation  by  the 
velocity  of  light,  or  186,300  miles  per  second.  The  same  result 
will  be  accomplished  if  in  the  second  member  R  is  expressed 
in  units  of  the  earth's  mean  distance  from  the  sun,  and  we 
introduce  the  time  required  by  light  to  cross  it.  This  is  called 
the  equation  of  light,  and  has  for  its  value  498.5  sec.  or  8.308 
min.  The  second  value  is  the  one  used  in  this  particular  case, 
since  the  predictions  are  not  carried  beyond  the  fraction  of  an 
hour. 

In  the  figure,  the  heliocentric  minimum  occurs  before  the 
geocentric,  hence  the  correction  must  be  subtracted  from  the 
latter  in  order  to  obtain  the  former,  which  is  the  one  sought. 
The  formula  thus  becomes  that  given  by  Hartwig,  — 

the  Heliocentric  time  =  the  Geocentric   time 

-  8m.308  R  cos  £  cos  (A,-  O). 

,  The  quantities  which  are  constant  for  each  star  are  fa  and  X, 
while  R  and  O>  the  co-ordinates  of  the  sun,  vary  with  the  time 
of  year.  Since  the  correction  is  only  necessary  for  stars  which 
change  very  rapidly,  and  the  maxima  or  minima  of  which  can 
be  determined  with  great  accuracy,  it  applies  chiefly  to  vari- 
ables of  the  Algol  type,  to  those  of  the  8  Cephei  type,  which 
have  very  short  periods,  and  to  the  /3  Lyrae  type.  For  the 
convenience  of  the  observer,  Hartwig  has  included  in  the 
Ephemeriden  for  1914  for  these  stars,  the  value  of  X  for  1900, 
and  also  log  8m.308  cos  /3,  so  that  the  necessary  correction  can 
be  computed  with  readiness. 


CHAPTER  XII 

ECLIPSING  BINARIES 

SINCE  several  types  of  variable  stars  are  spectroscopic  bina- 
ries, it  seems  desirable  to  the  writer  to  discuss  at  considerable 
length  the  principles  underlying  the  determination  of  motion 
in  the  line  of  sight,  for  it  is  this  motion  which  demonstrates 
the  binary  character  of  the  stars  and  furnishes  important 
material  regarding  their  orbits. 

Wave-length  and  vibration  frequency  have  been  defined  in 
the  first  chapter,  but  in  order  to  bring  them  again  to  mind  the 
definitions  will  be  repeated  here. 

The  wave-length  is  the  distance  the  disturbance  in  the  ether 
has  traveled  while  the  original  particle  is  executing  one  vibra- 
tion, =  X, 

The  vibration  frequency  is  the  number  of  vibrations  per- 
formed by  a  particle  in  one  second,  =  n, 

The  velocity  of  light  is  the  distance  traveled  by  light  during 
one  second,  =  V; 

Hence  the  following  relation  exists  between  n,  V,  and  X, 

F 
W=V 

n  is  thus  the  number  of  vibrations  which  fall  upon  the  eye 
during  one  second.  The  numerical  value  of  n  depends  upon 
the  wave-length  of  the  vibration  emitted  by  the  source  of  light. 
If  anything  were  to  happen  to  change  the  number  of  vibrations 
falling  upon  the  eye  during  a  second,  the  result  would  be  to 
change  the  effective  wave-length  without  in  reality  altering  it 
at  the  source.  Such  a  result  could  easily  be  produced  if  the 
source  of  light  were  to  move  toward  or  away  from  the  observer 
at  an  extremely  rapid  rate,  say  several  kilometers  per  second, 
and  similarly  if  the  observer  were  to  move  rapidly  to  and  fro. 
What  actually  happens  is  that  more  or  fewer  vibrations  fall 


ECLIPSING  BINARIES  217 

upon  the  eye  per  second,  and  consequently  the  apparent  wave- 
length of  the  light  emitted  is  changed. 

This  phenomenon  may  be  further  described  as  follows:  The 
line  connecting  the  observer  with  the  source  of  light  is  called 
the  line  of  sight.  If  the  source  is  moving  rapidly  toward  the 
observer,  many  more  vibrations  than  usual  will  fall  upon  the 
eye  per  second  and  the  effect  will  be  to  shorten  all  of  the  wave- 
lengths. If  the  spectrum  is  continuous,  no  change  in  it  will  be 
perceived,  because  if  a  wave-length  in  the  red  is  shortened  and 
its  position  shifted,  the  one  adjacent  to  it  is  also  shortened  and 
moves  up,  as  it  were,  to  take  the  vacant  place.  On  the  other 
hand,  if  there  is  an  absorption  spectrum  which  consists  of  the 
continuous  background  crossed  by  dark  lines,  then  a  change  is 
perceptible,  for  while  the  continuous  spectrum  remains  un- 
changed, the  dark  lines  are  all  shifted  toward  the  violet  and  as 
they  are  isolated  from  one  another,  there  is  no  adjacent  line 
moving  up  to  take  the  vacant  place.  Vice  versa,  if  the  source  of 
light  is  moving  rapidly  away  from  the  observer,  the  lines  will 
all  be  shifted  toward  the  red  end  of  the  spectrum.  Motion 
toward  the  observer  is  usually  called  approach,  and  motion 
away  recession,  and  the  rate  at  which  the  body  is  moving  is 
called  its  ra^iajj^elogity,  or  velocity  in  the  line  of  sight.  The 
statement  is  also  often  made  in  this  form:  approach  shortens 
the  wave-lengths  and  causes  the  lines  to  shift  toward  the  violet 
end  of  the  spectrum,  while  recession  increases  the  wave-lengths 
and  shifts  the  lines  toward  the  red  end  of  the  spectrum. 

The  first  knowledge  of  these  very  interesting  and  important 
facts  resulted  from  the  investigations  of  two  physicists,  Doppler 
and  Fizeau.  The  former,  in  1842,  announced  that  rapid  ap- 
proach or  recession  would  cause  a  change  in  the  wave-length, 
but  he  thought  erroneously  that  the  color  of  the  star  would  be 
changed,  ignoring  the  fact  just  mentioned  that  all  the  wave- 
lengths were  shifted  at  the  same  time,  and  hence  there  would 
be  no  change  in  the  color.  Fizeau,  in  1848,  was  the  first  to 
announce  the  facts  correctly,  viz.,  that  the  background  of  the 
continuous  spectrum  remains  unchanged,  while  the  dark  lines 


218         THE  STUDY  OF  VARIABLE  STARS 

shift  back  and  forth  upon  it.  He  outlined  the  method  of  meas- 
uring the  motion  of  a  heavenly  body  in  the  line  of  sight  by 
means  of  the  displacement  of  the  lines  in  its  spectrum,  but  not 
in  such  a  way  as  to  be  of  practical  use  to  astronomers.  His 
results  were  not  even  published  until  1870.  The  principle  is 
now  known  as  the  Doppler-Fizeau  principle. 

In  order  to  discover  whether  the  lines  in  the  spectrum  of  a 
star  are  displaced,  it  is  necessary  to  compare  them  with  lines 
coming  from  a  source  of  light  that  is  stationary  with  reference 
to  the  observer.  Such  a  source  can  be  found  in  some  terrestrial 
substance,  which  is  called  a  comparison  spectrum  when  it  is 
used  for  this  purpose,  or  is  sometimes  known  as  a  normal  spec- 
trum. The  history  of  the  application  of  spectroscopic  measure- 
ments to  celestial  bodies  is  full  of  interest.  The  first  astronomer 
actually  to  attempt  it  was  Sir  William  Huggins,  in  1862  and 
1863.  Since  the  name  of  this  brilliant  and  original  investigator 
is  linked  with  so  many  of  the  discoveries  in  astronomical  spec- 
troscopy  it  will  be  of  interest  to  the  reader  to  hear  his  own  ac- 
count of  the  beginning  of  his  work  in  this  branch  of  astronomy. 

In  1858,  having  equipped  an  observatory  with  very  excellent 
apparatus,  including  an  eight-inch  refractor  by  Alvan  Clark,  of 
Cambridgeport,  Massachusetts,  he  paused  to  consider  what 
problems  he  should  plan  to  investigate.  Many  years  later  he 
recalled  his  ideals  at  that  time  hi  the  following  words:1 

I  soon  became  a  little  dissatisfied  with  the  routine  character  of 
ordinary  astronomical  work,  and  in  a  vague  way  sought  about  in  my 
mind  for  the  possibility  of  research  upon  the  heavens  in  a  new  direc- 
tion or  by  new  methods.  It  was  just  at  this  time  .  .  .  that  the  news 
reached  me  of  Kirchoff's  great  discovery  of  the  true  nature  and  the 
chemical  constitution  of  the  sun  from  his  interpretation  of  the  Fraun- 
hofer  lines.  .  .  .  Here  at  last  presented  itself  the  very  order  of  work 
for  which  in  an  indefinite  way  I  was  looking  —  namely,  to  extend  his 
novel  methods  of  research  upon  the  sun  to  the  other  heavenly  bodies. 
A  feeling  as  of  inspiration  seized  me.  I  felt  as  if  I  had  it  now  in  my 
power  to  lift  a  veil  which  had  never  before  been  lifted;  as  if  a  key  had 
been  put  into  my  hands  which  would  unlock  a  door  which  had  been 

1  Nineteenth  Century  Review,  June,  1897. 


ECLIPSING  BINARIES 

regarded  as  forever  closed  to  man  —  the  veil  and  door  behind  which 
lay  the  unknown  mystery  of  the  true  nature  of  the  heavenly  bodies. 

His  initial  enthusiasm  never  deserted  him.  He  was  always 
entering  upon  new  experiments  in  connection  with  stellar  spec- 
troscopy,  and  his  name,  like  that  of  Herschel,  is  connected  with 
many  an  original  method  of  attack. 

Astronomers  in  other  countries  were  also  fascinated  by  the 
new  methods,  and  almost  simultaneously  investigations  were 
carried  on  by  Vogel  in  Germany,  Rutherford  in  New  York,  and 
Secchi  in  Rome.  However,  it  is  not  possible  to  give  the  histori- 
cal details  in  the  development  of  the  method  of  determining 
the  radial  velocity  of  a  star.  Many  mechanical  obstacles  had 
to  be  surmounted  and  the  observations  were  of  extreme  diffi- 
culty and  fatiguing  to  the  eye.  It  was  not  until  the  perfection 
of  the  photographic  dry  plate  that  it  was  possible  to  obtain  the 
sort  of  results  which  now  afford  such  unlimited  opportunity  for 
investigation.  It  is  true  that  with  a  large  telescope  like  that  of 
the  Lick  Observatory  accurate  measurements  of  the  displace- 
ments of  stellar  lines  could  be  made,  but  for  the  ordinary 
observer,  such  an  instrument  was  out  of  the  question.  Hence 
when  Vogel  and  Scheiner  showed  that  with  the  aid  of  the  photo- 
graphic plate,  the  displacement  of  lines  in  a  star  of  the  second 
or  third  magnitude  could  be  determined  with  a  twelve-inch 
telescope  with  as  much  accuracy  as  that  of  a  first-magnitude 
star  with  the  thirty-six-inch  telescope,  a  great  field  of  research 
was  opened  to  the  worker  with  an  average  instrument. 

In  order  to  understand  more  fully  the  method  of  measuring 
radial  velocity  at  the  present  time,  a  detailed  description  of  the 
Bruce  spectrograph1  of  the  Yerkes  Observatory  will  be  given. 
It  may  be  stated  in  passing  that  an  instrument  for  making 
visual  observations  of  the  spectrum  of  a  star  is  generally  called 
a  spectroscope,  while  one  which  photographs  it  exclusively  is 
called  a  spectrograph.  The  photograph  of  the  spectrum  is 
known  as  a  spectrogram. 

Two  views  of  the  Bruce  spectrograph  will  be  presented,  one 
1  E.  B.  Frost,  Ap.  J.t  15, 1. 


220          THE  STUDY  OF  VARIABLE  STARS 

in  which  it  is  attached  to  the  telescope,  and  the  other  showing 
the  interior  construction,  the  outer  covering  having  been 
removed.  At  the  ocular  end  of  the  great  telescope  is  a  large 
iron  ring  which  is  racked  in  or  out  according  to  the  kind  of  ap- 
paratus that  is  attached  to  the  instrument.  When  the  ordinary 
micrometer  is  in  use,  it  is  run  in  close  to  the  end  of  the  tube, 
but  when  the  spectrograph  is  wanted,  it  is  run  out  and  to  it  is 
attached  directly  the  ring  which  is  the  foundation  of  the  mount- 
ing of  the  spectrograph.  This  latter  is  completely  covered  with 
a  large  aluminium  case  which  serves  as  a  protection  against 
variation  in  temperature.  Its  walls  are  double  and  the  inter- 
vening air  space  is  filled  with  felt.  Inside  of  it  is  a  coil  of  wire 
which  becomes  heated  when  a  current  of  electricity  is  passed 
through  it,  thus  raising  the  temperature  of  the  spectrograph. 
A  thermometer  is  inserted  so  that  its  bulb  is  within  the  inner 
case,  and  any  change  in  temperature  indicated  can  be  corrected 
by  turning  on  the  electric  current,  the  purpose  being  to  keep 
the  temperature  as  constant  as  possible,  in  order  to  avoid  errors 
which  might  arise  from  unequal  expansion  of  the  different  parts 
of  the  instrument,  causing  a  displacement  of  the  prisms. 

From  the  large  ring  are  seen  projecting  three  tubes,  of  which 
the  shortest  one  is  centrally  placed  in  the  ring,  and  hence  in  the 
optical  axis  of  the  great  telescope.  It  carries  the  slit,  and  is  the 
outer  part  of  the  collimating  telescope  of  the  spectrograph.  To 
the  right  of  it  is  a  tube  of  the  shape  called  "gooseneck,"  which 
is  part  of  the  special  apparatus  for  guiding  the  instrument.  This 
guiding  must  be  done  with  great  accuracy  because  the  slit  of 
the  collimator  is  extremely  narrow,  and  the  star  image  formed 
at  the  focus  of  the  objective  very  small,  hence  there  will  be  con- 
siderable difficulty  in  keeping  the  image  on  the  slit.  The  ordi- 
nary clockwork  is  not  sufficient  for  this  purpose,  and  some 
method  must  be  devised  by  which  the  image  on  the  slit  can  be 
seen  and  watched  by  the  observer.  The  jaws  of  the  slit  are  of 
speculum  metal  which  is  susceptible  of  a  very  high  polish,  and 
they  are  inclined  at  a  very  slight  angle,  so  that  the  extra  light 
from  the  star  image  which  does  not  enter  the  slit  is  reflected 


ECLIPSING  BINARIES 

back  onto  a  pair  of  diagonal  prisms  in  the  gooseneck;  from 
here  by  other  reflections  they  are  passed  down  through  a  small 
telescope  enclosed  in  the  jacket  of  the  spectrograph,  and  are 
viewed  by  the  observer  through  the  eyepiece  which  projects 
from  its  lower  end.  He  is  able  by  electric  motors  to  control  the 
motion  of  the  clock,  and  thus  keep  the  image  on  the  slit.  The 
reader  will  understand  that  this  work  may  become  very  ardu- 
ous when  a  faint  star  or  a  nebula  is  under  observation  and  a 
long  exposure  of  seven  or  ten  hours  must  be  made.  It  would 
appear  from  the  photograph  that  the  gooseneck  obstructs  the 
path  of  the  ray  of  light,  but  there  is  an  aperture  through  which 
a  free  passage  is  allowed. 

The  tube  to  the  left  holds  the  apparatus  for  providing  the 
comparison  spectrum,  which  is  arranged  so  that  four  different 
substances  can  be  utilized.  Metallic  electrodes  are  employed 
and  the  spark  discharge  is  passed  between  them.  At  the  Yerkes 
Observatory  iron  and  titanium  are  chiefly  used,  and  helium  in 
a  vacuum  tube  can  take  the  place  of  one  of  the  pairs  of  elec- 
trodes. When  the  comparison  spectrum  is  being  formed,  a 
diaphragm  covers  the  central  portion  of  the  slit  where  the  star 
image  falls. 

The  plate  shows  also  a  small  tube  projecting  centrally  from 
the  main  tube  of  the  telescope.  This  contains  a  correcting  lens 
which  is  situated  about  twenty-one  inches  inside  of  the  tube. 
Such  a  lens  is  always  necessary  when  a  telescope  which  is 
intended  for  visual  work  is  used  for  photographic  purposes.  Its 
object  is  to  correct  for  the  chromatic  aberration  of  the  objec- 
tive, in  the  following  manner.  The  objective  cannot  bring  all 
of  the  different  wave-lengths  to  a  focus  at  the  same  distance 
behind  it.  When  it  is  to  be  used  for  visual  work  it  is  ground  to  a 
curvature  which  will  bring  together  the  wave-lengths  to  which 
the  eye  is  most  sensitive,  which  are  in  the  orange,  yellow,  and 
green.  The  focal  point  for  the  blue  and  violet  rays  is  some 
distance  inside  this  point.  But  these  rays  are  the  very  ones  that 
are  most  active  photographically,  hence  they  must  in  some 
way  be  brought  to  a  focus  together.  A  lens  which  is  used  exclu- 


222         THE  STUDY  OP  VARIABLE  STAES 

sively  for  photography  is  ground  for  it  in  the  first  place,  but  a 
visual  lens  may  be  changed  into  a  photographic  one  by  the  ad- 
dition of  a  small  correcting  lens  placed  within  the  focal  point, 
which  can  be  made  so  as  to  correct  for  any  desired  wave-length. 
(See  Fig.  21.) 

The  second  plate  shows  the  essential  parts  of  the  spectro- 
graph  itself,  and  it  is  to  be  noted  that,  with  a  few  differences, 
they  are  the  same  as  those  mentioned  in  describing  the  simple 
laboratory  spectroscope  in  Chapter  I,  viz.,  the  slit,  collimat- 
ing  telescope,  prism,  and  view  telescope.  It  has  three  prisms 
instead  of  one,  a  photographic  plate  holder  is  attached  to  the 
end  of  the  view  telescope  instead  of  an  eyepiece,  thereby  turn- 
ing it  into  a  camera,  and  there  are  two  additional  parts,  one  of 
which  is  the  apparatus  for  producing  the  comparison  spectrum, 
and  the  other  for  guiding  the  telescope.  Both  of  them  have 
been  referred  to  in  the  preceding  paragraphs. 

We  can  also  examine  on  this  plate  the  interior  of  the  spectro- 
graph.  Projecting  outside  of  the  heavy  ring,  which  is  its  main 
support,  can  be  seen  the  slit  tube,  the  gooseneck,  and  the  com- 
parison apparatus.  The  gooseneck,  with  its  tube,  can  be  fol- 
lowed until  the  end  is  reached,  where  the  eye  of  the  observer  is 
placed.  The  slit  is  in  the  outer  end  of  the  collimating  telescope, 
the  object  end  of  which  faces  inward  toward  the  first  prism. 
Over  the  third  prism  is  placed  the  view  telescope,  which  is  con- 
verted into  a  camera,  having  a  plate  holder  at  the  other  end. 
The  prisms  are  adjusted  so  that  they  are  effective  for  the  wave- 
length 4500  A.U. 

The  other  tubes  and  rods  which  are  represented  in  the  plate 
are  for  the  purpose  of  making  the  entire  instrument  as  rigid  as 
possible.  It  rests  upon  a  carriage,  at  an  angle  which  makes  it 
most  convenient  for  attachment  to  the  telescope.  On  the  table 
under  it  is  a  second  camera  which  may  be  interchanged  with 
the  one  already  in  use.  The  three  essentials  for  a  good  spectro- 
graph  are  thus  obtained;  i.e.,  the  parts  are  rigidly  connected  so 
that  there  is  no  flexure  in  the  instrument,  the  temperature  can 
be  kept  constant,  and  the  optical  parts  have  been  carefully 


ECLIPSING  BINARIES 

selected.  Something  may  be  said  at  this  point  in  regard  to  the 
loss  suffered  by  the  light  of  a  star  in  passing  through  such  an 
instrument.  The  ray  of  light  that  falls  upon  the  surface  of  the 
large  objective  must  pass  through  its  two  lenses,  which  are  of 
considerable  thickness,  and  then  through  the  correcting  lens. 
They  next  fall  upon  the  slit,  where  on  account  of  the  waverings 
of  the  atmosphere  there  is  often  much  loss,  then  in  order  through 
the  lens  of  the  collimator,  the  three  thick  prisms,  and  finally 
the  lens  of  the  camera.  Without  taking  into  account  the  loss 
of  light  at  the  slit,  Professor  Frost  estimates  that  hardly  more 
than  ten  per  cent  of  the  incident  light  is  transmitted  to  the 
photographic  plate,  and  when  the  atmospheric  conditions  are 
bad,  not  much  more  than  one  per  cent. 

The  next  step  in  determining  radial  velocity  is  to  measure 
the  displacement  of  the  lines  in  the  spectrogram  thus  obtained. 
This  is  done  with  a  special  kind  of  micrometer  called  a  measur- 
ing machine.  The  measurement  consists  in  bisecting  the  line 
in  the  comparison  spectrum  and  then  the  corresponding  line  in 
the  stellar  spectrum.  Many  difficulties  present  themselves  in 
the  process,  only  a  few  of  which  can  be  mentioned  here.  Since 
the  comparison  spectrum  is  an  emission  spectrum  its  lines  are 
bright,  but  appear  dark  on  the  negative.  The  spectrum  of  the 
star,  on  the  other  hand,  is  an  absorption  spectrum;  its  lines  are 
dark  and  appear  bright  on  the  negative.  Consequently  there  is 
a  possible  source  of  error  in  measuring  first  a  dark  line  and  then 
a  bright  one.  Another  difficulty,  and  a  very  great  one,  arises 
from  the  fact  that  in  the  star  spectrum  the  dispersion  is  usually 
quite  small  and  each  line  is  in  reality  a  group  of  lines.  The  wave- 
length of  this  group  must  be  taken  as  a  whole,  hence  the  im- 
portance of  giving  it  the  correct  value.  This  is  the  point  which 
requires  the  greatest  skill  and  judgment  on  the  part  of  the 
measurer. 

The  spectrum  as  it  appears  on  the  photographic  plate  is  a 
linear  spectrum,  and  the  displacements  of  the  lines  are  measured 
in  microns  or  thousandths  of  a  millimeter.  These  values  must 
be  changed  into  wave-lengths,  and  then  into  radial  velocity. 


224          THE  STUDY  OF  VARIABLE  STARS 

The  principle  which  connects  the  radial  velocity  with  the  wave- 
length is  as  follows:  — 
Let       X  =  the  original  wave-length, 

X'  =  the  wave-length  of  the  displaced  line, 
dX  =  X'  —  \  =  the  change  hi  wave-length, 
v  =  the  change  in  velocity,  or  motion  in  line  of  sight, 
V  =  the  velocity  of  light; 
then     t>:F=X'-X:X, 

and       v  =  —  .  F. 

It  will  be  seen  from  this  equation  that  the  value  of  v  depends 
upon  the  ratio  of  -r—  >  since  V  is  constant,  hence  in  the  spectrum 

A* 

of  a  given  star,  the  displacements  will  vary  with  the  wave- 
length, and  will  be  greater  in  the  red  end  of  the  spectrum  than 
in  the  violet. 

There  is  also  a  formula  well  known  to  spectroscopists  for 
converting  the  linear  differences  into  wave-length,  called  the 
Hartmann-Cornu  formula.  By  means  of  it  the  velocity  is 
obtained  in  kilometers  per  second.  When  the  velocity  corre- 
sponding to  the  linear  displacement  has  been  deduced,  it  is  not 
yet  the  radial  velocity  of  the  star,  for  a  portion  of  it  is  due  to 
the  motion  of  the  earth  itself,  and  must  be  eliminated.  The 
earth  in  its  revolution  around  the  sun  has  an  orbital  velocity 
of  nineteen  miles  per  second.  This  may  be  resolved  into  com- 
ponents in  any  desired  direction,  and  that  one  which  is  in  the 
line  connecting  the  earth  and  the  star  can  be  obtained  by  an 
appropriate  formula,  and  applied  to  the  observed  motion  to 
obtain  the  true  radial  velocity  of  the  star. 

The  action  of  the  spectrograph  must  be  tested  from  time  to 
time  to  see  if  it  is  in  perfect  adjustment  throughout  and  is  giv- 
ing accurate  results.  This  is  done  by  obtaining  the  radial  veloc- 
ity of  the  moon  or  of  a  planet  such  as  Venus  or  Jupiter.  In  each 
case  the  spectrum  observed  is  that  of  reflected  sunlight,  which 
offers  not  quite  so  much  difficulty  in  deciding  upon  the  wave- 
lengths of  groups  of  lines  as  do  the  stellar  spectra,  for  the  solar 


ECLIPSING  BINARIES 

spectrum  has  been  very  carefully  measured  and  the  intensities 
of  the  different  lines  in  a  group  are  well  known  from  Rowland's l 
table.  However,  additional  observations  of  certain  stars  se- 
lected as  fundamental  should  be  made  from  time  to  time  in 
order  to  discover  whether  there  are  systematic  differences  in 
the  results  obtained  at  the  observatories  which  are  engaged 
in  this  work.  As  stated  just  previously,  each  line  in  the  stellar 
spectrum  is,  on  account  of  the  small  dispersion,  composed  of  a 
group  of  lines,  and  the  wave-length  assigned  to  it  when  of  the 
solar  type  is  determined  largely  by  reference  to  the  correspond- 
ing group  in  the  solar  spectrum  taken  under  similar  conditions. 
In  the  various  types  of  stellar  spectra,  different  lines  which 
compose  the  group  will  have  different  intensities  and  hence  will 
displace  the  center  of  gravity  of  the  group,  which  is  the  point 
set  on  by  the  micrometer.  Therefore  fundamental  stars  of 
unlike  types  should  be  selected  and  the  results  from  different 
observatories  carefully  compared  and  combined. 

The  number  of  observatories  which  are  co-operating  in  this 
work  and  their  wide  distribution  on  the  earth  may  be  judged 
from  a  report  which  recently  appeared  in  the  Astrophysical 
Journal,2  April,  1914.  A  letter  was  sent  out  by  E.  B.  Frost,  the 
editor  of  the  Journal,  to  those  engaged  in  the  study  of  radial 
velocity,  asking  for  information  in  regard  to  their  investigations 
of  spectroscopic  binaries,  in  order  to  prevent  unnecessary 
duplication.  Answers  were  received  from  the  following  obser- 
vatories, thirteen  in  all:  Allegheny  Observatory  of  the  Univer- 
sity of  Pittsburg;  Detroit  Observatory,  University  of  Michigan; 
Dominion  Astronomical  Observatory,  Ottawa;  Harvard  Col- 
lege Observatory;  Konigliche  Sternwarte,  Bonn;  Kgl.  Astro- 
physikalisches  Observatorium,  Potsdam;  Lick  Observatory; 
Mount  Wilson  Solar  Observatory;  Paris  Observatory;  Pulkowa 
Observatory;  Royal  Observatory,  Cape  of  Good  Hope;  Uni- 
versity Observatory,  Vienna;  Yerkes  Observatory. 

Assuming  then  that  several  spectrograms  of  a  star  have  been 

1  Henry  A.  Rowland,  Preliminary  Table  of  Solar  Spectrum  Wave-Lengths. 


THE  STUDY  OF  VARIABLE  STARS 

taken  and  measured  with  as  great  care  as  possible,  the  results 
should  be  compared  to  see  how  well  they  agree.  If  on  examina- 
tion they  are  found  to  differ  not  more  than  a  few  kilometers, 
the  average  is  taken  and  assumed  to  be  the  radial  velocity  of 
the  star.  If,  on  the  other  hand,  the  variation  is  large  and  is 
repeated  at  regular  intervals,  it  is  evident  that  the  velocity  in 
the  line  of  sight  is  variable,  and  periodically  so,  hence  the  star 
must  move  to  and  fro  with  regularity  in  the  line  of  sight.  There 
is  only  one  explanation  for  this  phenomenon,  namely,  that  the 
star  is  one  of  a  pair,  each  of  which  must  be  moving  in  an  orbit 
around  the  common  center  of  gravity,  and  hence  together  they 
form  the  two  components  of  what  is  called  a  spectroscopic 
binary. 

Since  we  have  been  considering  a  spectrum  which  has  only 
one  set  of  lines,  the  body  which  is  the  companion  in  the  binary 
system  must  be  very  much  fainter  in  order  that  its  lines  shall 
not  appear  in  the  spectrum.  It  is  not  necessarily  entirely  dark, 
but  only  one  or  two  magnitudes  photographically  fainter  than 
the  brighter  component.  As  Campbell1  states, 

The  fourth  magnitude  companion  of  a  second  magnitude  star  of  the 
same  spectral  type  would  scarcely  be  able  to  impress  itself  upon 
the  primary's  spectrum.  The  invisible  components  in  any,  and 
perhaps  all,  spectroscopic  binaries  might  be  conspicuous  stars  if 
they  stood  alone. 

However,  there  are  frequent  cases  in  which  the  spectrum 
shows  the  presence  of  two  stars,  in  which  case  the  lines  become 
double  and  then  single  alternately.  These  are  recognized  at 
once  as  indicating  the  binary  character  of  the  star.  More  than 
three  hundred  spectroscopic  binaries  of  both  kinds  are  known 
at  the  present  time,  about  seventy  of  which  have  had  their 
orbits  computed. 

While  it  is  impossible  in  this  volume  to  give  any  account  of 
the  theoretical  method  of  obtaining  the  elements  of  a  spectro- 
scopic binary  from  observations  of  its  radial  velocity,  it  is  desir- 
able to  mention  briefly  certain  facts  to  which  allusion  is  fre- 
1  Stellar  Motions,  278. 


ECLIPSING  BINARIES 

quently  made.  When  a  series  of  measurements  has  been  col- 
lected, the  period  of  the  variation  is  the  first  element  to  be 
determined.  After  this  has  been  found,  the  observations  are 
arranged  according  to  the  phase,  as  is  the  case  of  a  variable 
star,  then  plotted  and  a  smooth  curve  drawn  through  them.  If 
the  star  has  no  irregularities  in  its  motion  and  the  observations 
are  accurate,  they  should  lie  quite  close  to  the  curve,  which  is 
called  the  velocity  curve.  It  is  by  measurements  taken  from 
this  curve  that  the  elements  can  be  determined,  though  they 
are  not  just  the  same  as  those  which  can  be  found  for  an  ordi- 
nary visual  binary  star.  An  excellent  statement  of  them  has 
been  given  by  Campbell  in  his  volume  on  Stellar  Motions.1  ^ 
Passing  now  to  the  consideration  of  the  question  whether 
many  variable  stars  are  spectroscopic  binaries,  we  find  the  evi- 
dence very  conclusive,  particularly  with  regard  to  certain  types 
of  variables.  While  we  may  not  yet  have  proved  the  case  with 
regard  to  all  the  members  of  any  given  class,  this  is  because 
many  of  them  are  too  faint  for  spectroscopic  investigation  at 
present,  and  hence  the  statement  does  not  apply  to  those  for 
which  the  spectrum  has  not  been  studied.  With  these  excep- 
tions we  may  lay  down  the  general  proposition  that  all  of  the 
short  period  variables  are  spectroscopic  binaries.  On  the  other 
hand,  none  of  the  long  period  variables,  and  none  of  those  which 
are  irregular,  are  binaries.  In  every  case  the  period  of  the  veloc- 
ity curve  is  equal  to  the  period  of  light  variation,  so  that  the 
existence  of  a  connection  between  the  type  of  the  variation  and 
the  binary  character  of  the  star  is  definitely  recognized.  The 
nature  of  the  relation  is  not  so  easily  explained;  in  fact  it  is 
understood  with  certainty  in  only  one  type,  the  Algol  type, 
though  numerous  theories  have  been  advanced  to  explain  the 
Cepheid  type.  The  explanation  of  the  variation  of  the  Algol 
type  will  be  taken  up  first.  It  depends  upon  evidence  derived 
from  the  light  curve  as  well  as  from  spectroscopic  observations. 
Algol  is  supposed  to  be  a  binary  star,  one  component  of  which 
is  very  much  fainter  than  the  other.  The  plane  of  the  orbit  of 
1  Stellar  Motions,  246-47. 


228          THE  STUDY  OF  VARIABLE  STARS 

revolution  is  inclined  only  a  very  little  to  the  line  of  sight,  so 
that  an  eclipse  of  each  component  by  the  other  occurs  once  in 
every  revolution. 

Goodricke,  in  1783,  was  the  first  to  offer  this  explanation, 
placing  it  at  the  end  of  a  communication  to  the  Royal  Society 
in  which  he  gave  an  account  of  his  observations  of  Algol  and 
the  determination  of  the  period.  He  says : 1 

If  it  were  not  perhaps  too  early  to  hazard  even  a  conjecture  on  the 
cause  of  this  variation,  I  should  imagine  it  could  hardly  be  accounted 
for  otherwise  than  by  the  interposition  of  a  large  body  revolving 
around  Algol,  or  some  kind  of  motion  of  its  own  whereby  part  of  its 
body,  covered  with  spots  or  such  like  matter,  is  periodically  turned 
towards  the  earth;  but  the  intention  of  this  paper  is  to  communicate 
facts,  not  conjectures,  and  I  flatter  myself  that  the  former  are  remark- 
able enough  to  deserve  the  attention  and  farther  investigation  of 
astronomers. 

The  eclipse  theory  is  suggested  by  the  appearance  of  the  light 
curve,  which  has  been  shown  in  Figure  10  and  described  briefly 
in  Chapter  I.  A  fuller  description  of  it  is  desirable  in  this  con- 
nection. It  is  therefore  repeated  in  Figure  29,  where  numbers 
are  attached  at  different  points  to  show  the  connection  with 
the  other  figures  and  for  convenient  reference.  An  inspection 
of  the  numbers  at  the  side  of  the  figure  will  show  the  change  of 
magnitude  during  the  variation  of  light.  The  maximum  bright- 
ness of  the  star  is  maintained  at  a  constant  magnitude  with  a 
slight  variation  of  less  than  .1  mg.  During  the  principal  mini- 
mum it  loses  about  1.1  mg.  The  descending  and  ascending 
branches  of  the  minimum  are  symmetrical,  and  the  minimum 
lasts  but  a  short  time.  The  secondary  minimum,  while  lasting 
about  the  same  length  of  time  as  the  principal  minimum,  is 
very  shallow,  there  being  a  change  of  about  .06  mg.  The  inter- 
val of  time  from  1  to  3  is  called  the  duration  of  phase.  We  can 
see,  then,  how  this  light  curve  suggested  an  eclipse  of  a  bright 
star  by  a  dark  one.  The  duration  of  phase  occurs  while  the 
dark  star  is  passing  in  front  of  the  bright  star,  the  eclipse  being 
partial.  The  time  of  minimum  represents  the  instant  of  greatest 
1  Phil.  Trans.,  73,482. 


ECLIPSING  BINARIES  229 

eclipse,  while  the  times  1  and  3  represent  the  first  and  last  con- 
tacts. During  the  rest  of  the  period  the  entire  surface  of  the 
bright  star  is  presented  to  view;  light  comes  from  it  alone,  and 
the  magnitude  is  very  nearly  constant.  In  fact,  from  the  time 
of  Goodricke  until  1910  no  evidence  was  offered  that  the  light 
was  not  constant  at  maximum,  and  it  was  only  with  the  appli- 
cation of  the  selenium  cell,  with  its  extreme  sensitiveness  to 
variations  of  light,  that  it  was  possible  to  determine  a  change 
so  small  as  .06  mg. 

The  loss  of  light  during  the  time  of  principal  minimum,  1.1 
mg.,  is  by  Pogson's  rule  equivalent  to  64  per  cent;  therefore 
two  thirds  of  the  central  body  is  obscured  by  the  dark  compan- 
ion. Considering  the  duration  of  phase  in  relation  to  the  entire 
length  of  period,  the  angle  described  by  the  satellite  during  the 
time  of  phase  is  about  50°.  This,  in  conjunction  with  their  sizes, 
would  require  the  radius  of  the  orbit  to  be  very  small  in  propor- 
tion to  the  size  of  the  principal  star,  when  compared  with 
visual  binaries.  Therefore  Goodricke's  idea  did  not  meet  with 
favor,  because  it  required  a  system  in  which  the  stars  were  very 
large,  and  yet  extremely  close  together.  His  idea  was  later 
revived  by  Pickering,  who  showed  by  calculation  that  the 
change  in  brightness  while  undergoing  eclipse  could  be  due  to 
obscuration  caused  by  a  totally  dark  satellite  coming  between 
the  earth  and  the  bright  primary;  but  still  the  theory  was  not 
generally  accepted.  When  spectroscopic  apparatus  was  per- 
fected, so  that  the  displacement  of  the  lines  could  be  measured, 
Vogel  very  easily  proved  the  binary  character  of  the  star,  and 
in  the  years  1888  and  1889  collected  observations  from  which 
he  was  able  to  determine  the  following  facts  regarding  the 
system. 

Diameter  of  the  principal  star 1,700,000  km. 

Diameter  of  the  satellite 1,330,000  km. 

Distance  between  their  centers 5,180,000  km. 

Orbital  velocity  of  Algol 42  km. 

Orbital  velocity  of  the  satellite 89  km. 

Masses  of  the  two  bodies f  and  f  of  the  sun, 

or  in  the  ratio  2:1. 


230          THE  STUDY  OF  VARIABLE  STARS 

This  was  on  the  supposition  that  the  satellite  was  a  dark  body, 
but  after  the  discovery  by  Stebbins1  of  the  secondary  minimum 
the  companion  was  no  longer  regarded  as  being  totally  dark, 
and  new  elements  of  the  variation  were  derived  and  given  the 
following  values. 

Radius  of  Algol 1.00 

Radius  of  companion 1.14 

Distance  between  centers 4.77 

Inclination  of  orbit 82°.3 

Surface  intensity  of  Algol 1.00 

Surface  intensity  of  faint  hemisphere  of  companion 0.050 

Surface  intensity  of  bright  hemisphere  of  companion. . .  .0.088 

Limiting  density  of  Algol 18O 

Limiting  density  of  companion 12O 

The  unusual  dimensions  of  Vogel's  system  explain  why  astrono- 
mers were  loath  to  believe  in  its  stability.  The  radius  of  the 
satellite  is  fully  .78  of  the  primary,  the  distance  between  their 
centers  only  3.05  of  the  same  unit,  and  the  distance  between 
their  surfaces  1.27  of  it. 

The  following  diagrams  will  show  the  relation  between  the 
light  curve,  the  orbital  motion,  the  displacement  of  the  spectral 
lines,  and  the  velocity  curve  of  the  star.  The  numbers  in  the 
different  diagrams  represent  corresponding  instants  of  time. 
They  may  be  described  as  follows. 

I.  The  Light  Curve.  The  time  from  1  to  3  is  known  as  the 
duration  of  phase;  1  represents  the  beginning  of  the  phase; 
2  the  time  of  minimum  or  the  middle  of  the  phase;  3  the  end  of 
the  phase;  4  the  instant  of  time  half-way  between  the  principal 
minimum  and  the  secondary  minimum;  5  the  beginning  of  the 
phase  for  the  secondary  minimum;  6  the  middle  of  the  second- 
ary minimum;  7  the  end  of  this  phase;  8  the  point  midway 
between  the  secondary  minimum  and  the  next  following  prin- 
cipal minimum;  and  9,  which  corresponds  to  1,  is  the  beginning 
of  the  phase  again. 

II.  The  Relative  Orbit.  In  this  diagram  the  larger  star,  A9  is 
kept  stationary  at  the  center  and  the  orbit  is  described  by  B, 

i  Ap.  J.t  32,  213. 


ECLIPSING  BINARIES  231 

relative  to  A.  The  parallel  lines  which  enclose  the  principal 
star  are  the  lines  of  sight  to  the  earth.  On  the  orbit  of  B  posi- 
tion 1  represents  the  instant  when  B  has  begun  to  encroach 
upon  the  disc  of  A,  i.e.,  it  is  the  beginning  of  an  eclipse;  2  repre- 
sents the  middle  of  the  eclipse,  when  the  fainter  star  is  pro- 
jected upon  the  brighter  star;  3  is  the  end  of  the  eclipse.  These 
positions  belong  to  the  primary  minimum,  because  the  brighter 


0.10 
0.30 
0.50 
0.70 

030 

1. 10 
1.30 


Figure  29.  Diagram  I 

THE  LIGHT  CURVE 


star  is  the  one  which  is  obscured.  4,  which  is  90°  from  2,  repre- 
sents the  point  half-way  between  the  principal  minimum  and 
the  secondary  minimum;  at  5  the  star  B  has  begun  to  disappear 
behind  the  disc  A,  which  marks  the  beginning  of  the  secondary 
minimum;  6  is  the  middle  of  the  secondary  minimum,  and  7 
represents  the  time  when  B  has  almost  completely  reappeared 
from  behind  A,  8  represents  the  point  where  B  is  again  in 
quadrature  with  A.  The  correspondence  between  Figures  I 
and  II  is  very  plain. 


THE  STUDY  OF  VARIABLE  STARS 

III.  The  Real  Orbits.  In  this  figure  C  is  the  center  of  gravity 
and  the  center  of  motion  of  both  of  the  components  of  the  sys- 
tem. Ay  being  the  more  massive  star,  describes  the  smaller 
orbit,  and  B  describes  the  larger  orbit.  At  any  instant  they  lie 


Figure  30.  Diagram  II 

THE  RELATIVE  ORBIT 

in  a  straight  line  passing  through  the  center  of  gravity.  For 
our  purpose  it  is  not  necessary  to  indicate  all  of  the  positions 
which  are  marked  on  the  light  curve,  and  only  those  at  the  con- 
junctions and  quadratures  are  given.  The  motion  in  the  orbit 
is  supposed  to  be  anti-clockwise. 


ECLIPSING  BINARIES 


233 


IV.  Spectroscopic  Evidence.  In  this  diagram  the  displace- 
ments of  the  lines  in  the  spectrum  of  the  principal  star  are  given, 
at  the  times  represented  by  2,  4,  6,  and  8.  The  upper  line  repre- 
sents the  normal  position  of  the  lines.  Since  Algol  is  of  spectral 


Figure  31.  Diagram  III 

THE  REAL  ORBITS 

type  A,  the  four  lines  are  supposed  to  be  the  hydrogen  lines, 
H7-H?. 

V.  The  Velocity  Curve.1  To  obtain  it  the  radial  velocities,  or 
displacements  of  the  lines,  are  plotted  as  ordinates  with  the 

1  Frank  Schlesinger  and  R.  H.  Curtiss,  Publications  of  the  Allegheny  Observa- 
tory of  the  University  of  Pittsburg*  i,  31. 


234 


THE  STUDY  OP  VARIABLE  STARS 

H£  _  HS  Hy 


!    4- 
6 


8 


Figure  32.  Diagram  IV 

SPECTROSCOPIC  EVIDENCE 


times  as  abscissas,  and  a  smooth  curve  is  drawn  through  the 
points  thus  indicated,  forming  what  is  called  the  velocity  curve 
of  the  star. 


o.o 


0-4-        08 


U 


2.8  <k 


T/fU    Mtt. 

+  30 

/ 

"~N 

+£0 

/ 

\ 

tlO 

t 

* 

\ 

0       A     V 

\ 

^ 

2  V 

-10         * 

A 

\ 

-2.0 

\ 

\ 

-30 

\ 

/ 

-40 

v 

X 

A- 

Figure  33.  Diagram  V 

THE  VELOCITY  CURVE 


ECLIPSING  BINARIES  235 

It  now  remains  to  point  out  the  connection  between  these 
different  diagrams.  Comparing  first  I  and  II,  we  see  from  the 
correspondence  of  the  numbers  the  relative  positions  of  the 
primary  and  satellite  during  one  revolution.  As  soon  as  B 
reaches  point  1  the  light  begins  to  diminish,  and  reaches  a  mini- 
mum at  the  time  of  conjunction,  2.  It  then  begins  to  increase, 
and  at  3  B  is  ready  to  pass  from  in  front  of  A.  Since  the  orbit 
is  circular,  point  4  will  be  halfway  between  the  primary  and 
secondary  minima.  Since  A  is  the  brighter  star,  the  eclipse 
just  described  will  be  the  darker  one,  and  will  correspond  to  the 
principal  minimum.  At  position  5  the  darker  star,  B,  has 
begun  to  be  occulted  behind  A,  and  the  descent  to  the  sec- 
ondary minimum  has  begun.  6  is  the  position  where  B  is 
directly  behind  A,  and  represents  the  middle  of  the  second- 
ary minimum.  The  very  small  change  in  magnitude,  0.06  mg., 
indicates  that  B  is  very  faint  in  proportion  to  A.  7  represents 
the  instant  when  B  is  about  to  emerge  from  behind  A,  and  8 
again  is  the  point  midway  between  the  two  eclipses.  In  Dia- 
gram III  the  numbers  correspond  to  those  in  Diagram  II,  but 
in  this  case  A  also  describes  an  orbit  about  the  center  of 
gravity,  C,  showing  that  the  orbital  motion  of  A  is  to  be  con- 
nected directly  with  the  displacement  of  the  lines  in  IV,  be- 
cause A  is  the  bright  body,  and  is  the  only  one  which  gives  a 
spectrum. 

At  any  instant  the  motion  of  A  is  in  a  direction  tangent  to 
the  orbit  at  that  point.  In  position  2  its  motion  will  therefore 
be  at  right  angles  to  the  line  of  sight,  and  hence  the  lines  in  the 
spectrum  will  not  be  displaced,  but  will  be  in  their  normal  posi- 
tion. In  position  4,  90°  from  2,  the  motion  of  A  will  be  entirely 
in  the  line  of  sight,  and  directed  toward  the  observer;  therefore 
the  lines  will  have  their  maximum  displacement  toward  the 
violet  end  of  the  spectrum.  At  6  it  is  again  moving  across  the 
line  of  sight,  and  the  lines  will  be  in  their  normal  position,  while 
at  8  it  will  have  its  maximum  velocity  of  recession,  and  the  lines 
will  be  displaced  toward  the  red.  If,  therefore,  observations  of 
the  spectrum  of  Algol  show  that  at  the  time  of  the  principal 


236  THE  STUDY  OF  VARIABLE  STARS 

minimum  the  lines  are  in  their  normal  position,  are  then  dis- 
placed toward  the  violet,  regain  their  normal  position  at  the 
time  of  the  secondary  minimum,  and  then  swing  toward  the 
red  end  of  the  spectrum,  indicating  recession  from  the  earth, 
the  connection  between  the  orbital  motion  of  Algol  and  the 
displacement  of  the  lines  in  its  spectrum  is  complete. 

At  the  times  between  3  and  5,  and  7  and  1,  the  light  comes 
from  both  stars,  and  therefore  is  at  a  constant  maximum.  In 
the  velocity  curve  2  and  6  represent  the  normal  positions  of 
the  lines,  where  the  radial  velocity  is  0;  4  represents  the  maxi- 
mum positive  velocity,  or  greatest  rate  of  approach,  and  8  the 
greatest  negative  velocity,  or  greatest  rate  of  recession.  If  the 
system,  as  a  whole,  has  a  radial  velocity  independent  of  its 
orbital  motion,  this  will  be  indicated  in  the  velocity  curve  by 
the  fact  that  the  areas  of  the  curve  above  and  below  the  zero 
line  will  not  be  equal.  If  this  is  the  case,  the  velocity  of  the 
system  can  easily  be  found  by  drawing  an  abscissa  which 
shall  be  an  axis  of  symmetry.  This  is  indicated  by  the  dotted 
line. 

If  both  bodies  in  the  system  are  bright  there  will  be  two  sets 
of  lines  in  the  spectrum.  When  they  are  of  the  same  spectral 
type  the  two  spectra  will  be  identical,  except  that  perhaps  one 
may  be  fainter  than  the  other.  Also,  if  they  have  unequal 
masses  the  orbits  described  by  the  two  will  be  of  unequal  size, 
and  the  resulting  displacements  of  the  lines  will  not  be  the  same; 
therefore  the  more  massive  body  will  be  represented  by  the 
lines  having  the  smaller  displacement,  and  the  smaller  body 
by  the  lines  having  the  greater  displacement.  When  they  are 
moving  across  the  line  of  sight  the  spectrum  will  be  normal,  as 
before. 

We  now  come  to  a  consideration  of  the  various  combinations 
which  are  possible  in  an  eclipsing  system,  and  it  is  clear  that 
the  evidence  which  we  have  comes  from  two  independent 
sources,  as  follows:  (1)  the  character  of  the  curve,  including 
the  duration  of  phase;  and  (2)  the  character  of  the  spectrum. 
From  these  facts  we  must  decide  upon  the  physical  and  orbital 


03 


ECLIPSING  BINARIES  237 

relations  of  the  two  components.  The  light  curves  may  be  of 
four  kinds. 

(1)  There  may  be  a  series  of  equal  minima,  similar  to  each 
other  in  every  respect,  occurring  at  equal  intervals. 

(2)  There  may  be  a  series  of  equal  minima  occurring  at 
unequal  intervals,  but  arranged  in  pairs,  two  and  two,  the  alter- 
nate intervals  being  equal. 

(3)  There  may  be  a  series  of  unequal  minima  occurring  at 
equal  intervals,  the  alternate  minima  being  always  equal  and 
similar  to  each  other. 

(4)  There  may  be  a  series  of  unequal  minima  at  unequal 
intervals,  where  the  alternate  minima  are  equal  and  similar 
and  the  alternate  periods  are  equal. 

In  the  above  description  the  term  equal  minima  refers  to  the 
change  in  brightness  during  the  minimum,  and  to  the  shape  of 
the  curve.  The  character  of  the  duration  of  phase  also  forms  a 
fifth  source  of  evidence  which  may  be  expressed  as  follows, 

(5)  The  minimum  may  be  prolonged,  lasting  an  hour  or 
more,  or  it  may  be  very  brief,  not  over  twenty  minutes  in 
duration. 

The  spectrum  may  be  either  single  or  double,  i.e.,  there  may 
be  one  set  of  lines  or  two:  in  the  former  case  one  body  is  bright, 
and  the  other  either  dark  or  else  very  much  fainter;  in  the 
second  case  both  bodies  are  bright.  In  the  latter  the  two  spectra 
may  be  of  the  same  class,  in  which  case  all  of  the  lines  will  be 
doubled  at  the  time  of  greatest  displacement;  or  if  they  are  of 
different  types  only  certain  of  the  lines  will  be  doubled.  This  is 
illustrated  by  Plate  X,  which  shows  the  spectrum  of  Mizar,  or 
f  Ursae  Majoris,  taken  at  two  different  times.  In  one  case  the 
lines  are  single  and  in  the  other  they  are  double. 

Plate  XI  shows  the  spectrum  of  p  Orionis,  taken  on  two 
different  dates,  showing  a  change  in  the  displacement  of  the 
lines  and  hence  a  variable  radial  velocity. 

We  must  next  consider  the  theoretical  half  of  the  problem, 
that  is,  the  physical  relations  of  the  two  components  and  their 
orbital  movements.  Upon  the  former  depend  the  phases,  and 


238          THE  STUDY  OF  VARIABLE  STARS 

upon  the  latter  the  intervals  between  the  eclipses.  By  the 
physical  properties  of  the  two  bodies  are  meant  their  sizes  and 
brightness  per  unit  area,  called  by  Stebbins  surface  intensity, 
but  sometimes  known  as  intrinsic  brightness.  These  two  prop- 
erties determine  the  total  brightness  of  the  stars.  Four  different 
combinations  may  arise,  for  the  stars  may  be  of  equal  or  un- 
equal size  and  their  surface  intensities  may  be  the  same  or  not. 
It  is  only  possible  to  decide  which  combination  is  present  when 
there  are  two  minima  in  the  light  curve. 

Theoretically  the  orbits  may  be  circular  or  elliptical.  If  the 
latter,  the  major  axis  may  be  in  the  line  of  sight  or  inclined  to  it, 
and  the  plane  of  the  orbit  may  be  inclined  at  a  greater  or  less 
angle  to  the  line  of  sight.  It  is  obvious,  then,  that  there  are 
several  possible  combinations  of  stars  and  orbits  in  an  eclipsing 
system,  but  there  will  be  only  four  light  curves  to  represent 
them;  hence  some  curves  may  result  from  several  different 
combinations.  The  following  statements  will  indicate  most  of 
them.  The  numbers  refer  to  the  curves  described  a  few  pages 
earlier. 

(1)  This  curve  may  be  produced  by  a  system  in  which  there 
is  one  bright  body  and  one  dark  body,  or  two  bright  bodies  of 
equal  size  and  luminosity  or  surface  intensity.  The  orbit  may 
be  circular,  or  it  may  be  elliptical,  having  its  major  axis  coin- 
ciding with  the  line  of  sight.  It  will  not  be  possible  from  the 
light  curve  alone  to  decide  which  of  these  systems  is  the  correct 
one,  though  perhaps  it  may  be  judged  which  one  is  the  most 
probable.  Spectroscopic  observations  combined  with  the  evi- 
dence from  the  curve  will  settle  the  point,  for  if  there  is  only 
one  bright  body  there  will  be  but  one  set  of  lines,  whereas  if 
both  are  bright  they  must  be  equally  bright  in  order  to  produce 
equal  eclipses,  hence  each  will  produce  a  spectrum,  and  there 
will  be  two  sets  of  lines.  Measurements  of  the  velocity  curve 
will  determine  the  eccentricity  of  the  orbit  and  thus  decide 
whether  it  is  elliptical  or  circular.  If  it  is  elliptical,  then  the 
major  axis  must  be  in  the  line  of  sight,  or  else  the  eclipses  would 
not  occur  at  equal  intervals.  Unfortunately  this  evidence  is 


ECLIPSING  BINARIES  239 

lacking  for  many  of  the  Algol  variables  because  they  are  faint 
and  their  spectra  have  not  been  investigated,  therefore  we  are 
forced  to  confine  ourselves  to  data  derived  from  the  curve  alone. 

(2)  This  curve  arises  from  only  one  combination.  There  are 
two  bodies  equal  in  size  and  brightness  in  the  system,  the  orbit 
is  elliptical,  and  the  major  axis  is  not  in  the  line  of  sight. 

(3)  This  curve  may  result  from  either  of  two  combinations. 
There  must  in  any  case  be  two  unequally  bright  stars,  but  the 
orbit  may  be  circular,  or  it  may  be  elliptical,  with  the  major 
axis  in  the  line  of  sight. 

(4)  This  curve  also  requires  two  unequally  bright  stars,  but 
only  one  orbit  is  possible,  which  is  elliptical,  with  the  major 
axis,  making  an  angle  with  the  line  of  sight. 

The  fifth  point,  the  duration  of  minimum,  has  been  treated 
very  fully  by  Russell,1  who  has  written  a  series  of  very  impor- 
tant articles  dealing  with  the  mathematical  theory  of  deducing 
certain  elements  of  a  system  from  its  light  curve.  When  the 
minimum  is  very  short,  as  with  Algol,  the  eclipse  is  only  partial. 
When  it  remains  constant  for  some  time,  as  in  the  case  of  U 
Cephei  (see  Fig.  11),  the  eclipse  is  either  total  or  annular. 

The  difference  between  the  two  cases  can  readily  be  under- 
stood by  analogy  with  solar  eclipses,  save  that  there  is  a  greater 
disparity  in  the  sizes  of  the  two  bodies  concerned.  The  eclipse 
is  total  if  a  small  star  is  obscured  by  a  large  one,  in  which  case 
the  beginning  and  end  of  the  minimum  magnitude  correspond 
to  the  beginning  and  end  of  totality,  or  to  the  times  of  second 
and  third  contact.  Whereas  if  the  larger  star  is  eclipsed  by  the 
smaller  one  the  eclipse  is  annular  and  the  minimum  phase  is 
reached  as  soon  as  the  disc  of  the  smaller  star  is  entirely  pro- 
jected on  to  the  disc  of  the  larger  one.  The  times,  as  before,  of 
the  beginning  and  end  of  the  minimum  phase  will  correspond 
to  the  times  of  the  second  and  third  contacts.  Whether  in  any 
given  case  a  total  or  an  annular  eclipse  occurs  can  be  deter- 
mined only  by  trial,  after  we  have  found  the  relative  sizes  of 
the  two  stars  and  their  surface  intensities. 
1  Ap.  J.t  35,  315;  36,  54. 


240          THE  STUDY  OF  VARIABLE  STARS 

In  studying  the  light  curves,  Russell  and  Shapley1  have  also 
introduced  the  hypothesis  that  the  stars  may  be  darkened 
toward  the  limb  just  as  the  sun  is,  or  perhaps  even  to  a  greater 
extent.  Thus  for  each  star  of  this  type  the  orbit  of  which  has 
been  determined,  two  solutions  have  been  made,  called  the 
"uniform"  and  the  "darkened,"  the  first  on  the  assumption 
that  the  star's  light  is  uniformly  distributed  over  its  entire 
surface,  and  the  second  on  the  assumption  that  the  disc  is 
darkened  to  zero  at  the  edge. 

But  it  is  not  possible  to  carry  this  discussion  further  in  a 
theoretical  direction,  for  space  must  be  given  rather  to  some  of 
the  results  which  have  been  determined  by  Russell  and  Shapley 
in  applying  their  methods  to  the  examination  of  stars  of  this 
type.  The  following  are  some  of  the  facts  which  have  been 
gathered  from  their  various  papers. 

In  certain  cases,  the  dark  companion  has  a  volume  ten  times 
the  volume  of  the  brighter,  and  yet  scarcely  one  tenth  of  the 
total  light,  for  example,  S  Cancri.  In  some  cases  both  stars  are 
very  nearly  alike  in  all  respects,  as  ft  Aurigae.  Others  again 
are  similar  in  size  but  different  in  brightness,  as  U  Pegasi.  In 
the  majority  of  instances  the  dark  companions  are  larger  than 
the  bright  primaries.  Where  they  are  smaller,  the  difference  in 
size  is  always  slight. 

In  most  of  the  cases  investigated  by  Shapley  he  found  that 
the  fainter  star  was  self-luminous,  and  that  it  was  never  nec- 
essary to  assume  that  the  companion  was  entirely  dark.  He 
makes  the  further  statement2  that  in  about  two  thirds  of  the 
systems,  the  difference  in  brightness  of  the  two  components 
does  not  exceed  two  magnitudes,  and  that  no  observed  differ- 
ence is  greater  than  four  magnitudes.  It  will  be  seen  from  these 
facts  that  the  differences  are  not  greater  than  are  usually  found 
in  visual  binaries.  The  eclipse  of  a  bright  star  by  a  dark  com- 
panion of  much  less  than  one  half  of  its  radius  would  ordinarily 
escape  detection.  Where  it  has  been  possible  to  determine  the 

1  Pop.  AsL,  20,  572;  Ap.  J.t  36,  239,  385. 
«  Ap.  J.,  38,  172. 


ECLIPSING   BINARIES  241 

difference  in  color  at  the  time  of  eclipse,  the  large  faint  star  has 
been  found  to  be  the  redder.  Of  the  ninety  stars  investigated 
by  Shapley,  thirteen  have  spectral  type  B,  fifty  two  A,  ten  F, 
seven  G,  and  one  K,  six  are  marked  too  faint  for  observation 
and  one  is  left  blank. 

The  inclination  of  the  orbits  must  of  necessity  be  small 
in  order  to  secure  an  eclipse,  but  in  a  few  cases  of  partial 
eclipse  it  has  been  found  to  be  over  30°.  For  RR  Centauri 
it  is  48°. 

The  densities  of  the  stars  in  these  systems  are  on  the  whole 
much  less  than  that  of  the  sun,  though  there  are  some  excep- 
tions. 

Attention  should  be  called  at  this  point  to  an  important  piece 
of  research  work  just  published  by  Shapley1  called  "A  Study  of 
the  Orbits  of  Eclipsing  Binaries."  Unfortunately  it  comes  too 
late  for  the  writer  to  cull  from  it  anything  suitable  to  the  present 
chapter.  However,  some  of  the  material  in  it  has  already  been 
published  in  earlier  articles  in  the  periodicals. 

A  study  of  Shapley's  orbits  shows  that  he  has  included  /8 
Lyrae  and  similar  stars  with  those  of  the  strictly  Algol  type. 
This  is  not  an  unexpected  classification,  since  /3  has  long  been 
known  to  be  an  eclipsing  star,  the  difference  being  that  the  two 
components  are  considerably  flattened,  and  are  very  close 
together,  so  that  between  eclipses,  the  area  of  the  light-giving 
surface  turned  toward  us  is  not  uniform,  and  hence  the  light 
curve  does  not  give  a  constant  magnitude  at  this  time.  Twelve 
stars  placed  in  this  class  by  Hartwig  have  their  elements  stated 
among  the  ninety  of  Shapley.  The  accompanying  figure2  shows 
the  chief  characteristics  of  the  system.  The  peculiarities  of  the 
spectrum  will  be  described  later. 

It  would  appear  that  there  is  no  dividing  line  between  the 
members  of  the  two  groups  of  eclipsing  stars;  that  in  the  order 
of  evolution,  the  two  components,  as  they  are  first  formed  from 
the  original  nebulous  mass,  are  possibly  surrounded  by  the 

1  Contributions  from  the  Princeton  Observatory,  no.  3. 
8  G.  W.  Myers,  Ap.  J.t  7,  3. 


242          THE  STUDY  OF  VARIABLE  STARS 

same  gaseous  envelope,  and  because  of  their  small  density  and 
rapid  rotation  become  elliptical  in  shape.  In  the  course  of  time, 
owing  to  tidal  friction,  they  tend  to  separate,  and  get  farther 
apart  so  that  they  are  no  longer  in  contact.  There  will  always 
be  two  eclipses,  but  while  the  two  bodies  are  in  such  close 
proximity  to  each  other,  the  light  curve  will  vary  continuously. 
After  they  have  become  spherical  and  are  separated,  a  certain 


Figure  34 

THE  SYSTEM  OF  ft  LYRAE 

distance,  the  light  between  the  two  times  of  eclipse  will  come 
unimpeded  from  both  and  will  be  constant.  At  the  same  time 
it  is  not  at  all  certain  that  a  star  of  one  type  will  ultimately 
develop  into  one  of  the  other. 

A  discussion  of  /3  Lyrae  would  not  be  complete  without  an 
account  of  its  remarkable  spectrum,  which  is  undoubtedly  the 
most  complicated  and  interesting  one  in  the  sky.  It  consists  of 
bright  and  dark  lines  which  shift  their  positions  with  reference 
to  the  normal,  showing  quite  conclusively  that  it  is  a  binary 
and  that  somewhere  about  the  system  is  an  atmosphere  which 


ECLIPSING  BINARIES  243 

is  of  a  high  enough  temperature  to  give  emission  instead  of 
absorption  lines. 

This  fact  was  announced  by  Pickering1  in  1891  as  one  of  the 
first  fruits  of  the  Henry  Draper  Memorial  work  on  stellar  spec- 
tra. The  first  sentence  in  his  article  states :  "  The  spectrum  of 
the  variable  star  0  Lyrae  is  unlike  that  of  any  other  star  hith- 
erto examined."  He  next  calls  attention  to  the  swinging  back 
and  forth  of  the  bright  lines  which  was  coincident  with  certain 
phases  in  the  light  curve,  and  concludes  by  suggesting  that  this 
may  be  due  to  the  revolution  of  the  body  emitting  them,  and 
that  the  star  is  a  spectroscopic  binary  like  ft  Aurigae,  although 
he  offers  other  explanations  also,  ft  Lyrae  immediately  became 
an  object  of  study  to  spectroscopists,  but  in  spite  of  all  the 
labor  which  has  been  expended  on  it,  there  are  certain  points 
about  it  which  are  still  baffling.  The  most  recent  exhaustive 
study  of  it  was  made  at  the  Allegheny  Observatory  by  R.  H. 
Curtiss2  and  published  in  1911,  and  from  this  paper  a  few  of  the 
more  easily  understood  points  will  be  taken. 

As  stated  above,  the  spectrum  consists  of  dark  lines  and 
bright  lines,  that  is,  an  absorption  spectrum  and  an  emission 
spectrum.  The  dark  lines  are  clearly  identified  as  belonging  to 
two  separate  spectra,  one  of  which  is  of  type  B8A  and  the  other 
is  B5A.  The  first  set  of  lines  oscillates  with  a  range  of  369  km. 
in  the  period  of  the  light  variation.  The  second  set  is  appar- 
ently fixed  within  quite  narrow  limits.  The  lines  of  the  emission 
spectrum,  in  the  form  of  broad  bands,  accompany  nearly  all 
the  hydrogen  and  helium  lines  within  the  region  studied,  and 
do  not  oscillate,  but  none  exist  alone.  Narrower  emission  lines 
accompany  many  of  the  other  dark  lines.  The  hydrogen  and 
helium  lines  are  nearly  all  very  complex,  since  they  result  from 
a  combination  of  lines  in  all  three  spectra.  The  study  of  these 
lines  was  difficult  in  the  extreme,  and  they  could  not  in  general 
be  used  for  the  measurement  of  radial  velocity.  From  the  single 
dark  lines  a  sufficient  number  was  selected  upon  which  to  base 
the  measurement  of  the  radial  velocity.  There  is  not  much 

1  A.N.  3051.  2  Publications  of  the  Allegheny  Observatory,  xx,  73. 


244          THE  STUDY  OF  VARIABLE  STARS 

doubt  that  some  of  the  phenomena  of  the  complex  lines  are  due 
to  differences  in  pressure,  or  reversal,  or  other  physical  condi- 
tions under  which  the  atmospheres  of  the  component  stars 
exist.  The  special  attempts  to  study  and  measure  the  emission 
lines  resulted  only  in  the  conclusion  that  they  may  oscillate  in 
a  complex  manner,  but  that  more  probably  they  remain  fixed 
in  position  while  the  distribution  of  the  intensity  of  their  differ- 
ent parts  is  altered.  It  was  hence  impossible  to  measure  their 
positions  with  the  accuracy  desired  in  radial  velocity  measures. 

As  an  outcome  of  the  observed  results,  Curtiss  describes  two 
different  hypothetical  systems  which  may  explain  the  spectral 
changes  in  the  star.  It  seems  best,  however,  not  to  attempt  an 
explanation  in  this  place,  and  to  wait  until  further  evidence 
has  been  collected,  however  loath  one  may  be  to  leave  the  study 
of  this  fascinating  subject.  Undoubtedly  a  higher  dispersion 
such  as  may  be  obtained  with  instruments  like  the  Mt.  Wilson 
reflector  will  make  clearer  the  relation  of  the  different  parts  of 
the  complex  lines.  If  it  is  possible  to  isolate  the  central  point 
of  the  bright  lines  and  measure  their  displacements,  it  will 
assist  much  in  the  desired  result.  As  for  the  second  set  of 
dark  lines  which  do  not  oscillate,  they  are  considered  to  be  due 
not  to  a  third  body,  but  more  than  likely  to  reversals  in  the 
atmosphere, 

There  still  remains  one  class  of  variables  which  are  spectro- 
scopic  binaries,  namely,  the  Cepheid-Geminid  group.  All  of  the 
stars  of  these  two  groups  which  are  bright  enough  to  have  had 
their  radial  velocity  measured  are  spectroscopic  binaries,  and 
the  period  of  light  variation  agrees  in  every  case  with  that  of  the 
shifting  of  the  lines.  They  are  of  spectral  types  F  and  G.  The 
elements  of  the  orbits  of  twelve  of  these  stars  were  studied  and 
compared  by  Miss  Psyche  Sutton1  with  the  following  results. 

In  every  case  investigated,  only  one  set  of  lines  appears  in 

the  spectrum,  hence  only  one  of  the  component  stars  is  bright. 

The  eccentricities  are  large,  ranging  from  .10  to  .49,  and  the 

size  of  the  orbit  is  small,  that  is,  they  are  close  stars.  The  shape 

»  Pop.  Ast.t  19,  408. 


ECLIPSING  BINARIES  245 

of  the  light  curve  precludes  the  possibility  of  an  eclipse.  There 
are  some  variations  in  the  spectral  lines  not  due  to  orbital  mo- 
tion, and  there  is  also  a  significant  shifting  of  the  point  of  maxi- 
mum energy  in  the  spectrum  accompanying  the  light  change. 
A  still  more  important  fact  is  that  the  maximum  brightness 
occurs  very  nearly  at  the  time  when  the  primary  star  is 
approaching  the  observer  most  rapidly,  and  the  minimum  when 
it  is  receding  most  rapidly.  There  is  no  connection  between 
the  time  of  maximum  light  and  that  of  periastron  passage. 

Several  theories  have  been  offered  in  explanation  of  these 
facts,  of  which  only  four  will  be  mentioned.  The  first  is  that 
the  light  variation  is  due  to  tidal  action.  The  brighter  compo- 
nent is  the  satellite  and  the  dark  one  the  primary.  Since  the 
orbit  has  a  considerable  eccentricity,  when  the  bright  star  is  at 
periastron  it  is  much  nearer  the  primary  than  at  other  times, 
and  the  gravitational  force  would  hence  cause  enormous  tides, 
which  would  elongate  the  disc,  causing  a  greater  light-giving 
surface  to  be  presented.  Allowing  for  the  delay  in  the  crest  of 
the  tidal  wave  due  to  friction,  it  would  still  be  necessary  for  the 
maximum  brightness  to  occur  in  the  vicinity  of  periastron;  but 
this  relation  does  not  exist,  that  is,  there  is  no  connection 
between  the  time  of  maximum  and  the  time  of  periastron  pas- 
sage. Besides,  as  Miss  Clerke  suggests,  such  enormous  tides 
would  probably  disrupt  the  surface  and  cause  outbursts  of 
heated  gas,  which  would  be  indicated  by  the  presence  of  bright 
lines  in  the  spectrum,  and  furthermore  they  would  hardly  sub- 
side in  the  length  of  the  period,  which  is  after  all  quite  short  for 
the  action  of  such  great  forces.  However,  while  tidal  action  is 
not  the  main  force  acting  to  produce  the  variation  of  the  star, 
there  is  no  doubt  that  its  effect  is  felt  in  the  light  curve. 

Another  theory  advocated  by  Curtiss  is  that  the  system  is 
pervaded  by  a  resisting  medium  which  enhances  the  brightness 
of  that  side  of  the  star  that  faces  the  direction  of  motion.  Here 
again  the  brighter  star  is  the  satellite.  There  are  several  objec- 
tions to  this  theory,  one  being  that  the  resisting  medium  would 
have  to  be  rather  dense  in  order  to  produce  the  necessary  effect, 


246          THE  STUDY  OF  VARIABLE  STARS 

and  that  the  material  composing  it  would  in  time  diminish  as 
it  is  taken  up  by  the  brighter  star.  As  it  becomes  less  dense, 
the  effect  would  be  less  and  the  variation  of  the  star  in  bright- 
ness be  less.  Also  there  would  be  a  greater  range  in  brightness 
for  stars  having  more  eccentric  orbits,  which  has  not  proved  to 
be  the  case. 

Perhaps  the  most  acceptable  theory,  and  the  one  which  best 
fits  the  observed  connection  between  the  times  of  maximum 
brightness  and  maximum  velocity  of  approach,  is  one  proposed 
by  Duncan.1  He  supposes  that  the  brighter  star  is  the  satellite, 
and  that  the  entire  orbit  is  filled  with  a  very  rare  envelope  of 
nebulous  matter,  resembling  in  nature  the  corona  of  the  sun, 
but  much  less  dense  than  is  required  by  the  preceding  theory. 
As  the  star  is  carried  through  this  by  its  orbital  motion,  its 
atmosphere  is  brushed  back  by  the  friction  of  this  medium,  its 
depth  becomes  less  on  the  forward  side,  and  the  light  from  the 
photosphere  shines  out  more  brilliantly  since  it  passes  through 
a  much  smaller  layer  of  the  cooler  and  absorbing  atmosphere. 
This  would  agree  quite  satisfactorily  with  observation,  for 
when  the  star  is  approaching  the  observer  most  rapidly,  the 
layer  of  atmosphere  facing  us  would  be  at  its  thinnest,  and  the 
star  would  have  its  maximum  brightness.  On  the  other  hand, 
when  the  star  is  receding  most  rapidly,  the  thickest  layer  of  the 
atmosphere  is  turned  toward  us  and  the  star  is  at  its  minimum. 
This  theory  is  illustrated  by  the  accompanying  figure. 

The  fourth  theory  can  only  be  alluded  to  superficially.  Very 
recently  Shapley2  has  published  a  paper  in  which  he  states  that 
there  are  so  many  objections  to  each  of  the  theories  based  on  the 
binary  star  explanation  of  the  Cepheid  type  of  variation,  that 
it  seemed  better  to  him  to  reject  it  altogether  and  consider  the 
light  variation  as  being  due  to  some  intrinsic  change.  The  most 
promising  explanation  in  his  opinion  is  founded  on  the  concep- 
tion of  periodic  pulsations  in  the  masses  of  isolated  stars. 

Little  need  be  said  of  the  cluster  type  except  that  it  is  essen- 
tially identical  with  the  Cepheid  type,  the  division  being  usually 
1  L.O.B.,  151.  2  Ap.  J.,  40, 105. 


ECLIPSING   BINARIES  247 

based  on  length  of  period.   Shapley  suggests  that  Cepheids  of 
periods  less  than  a  day  shall  be  called  arbitrarily  cluster  type 


Figure  35 

THEORETICAL  SYSTEM  OF  8  CEPHEI 

variables,  for  there  is  at  present  no  evidence  of  real  difference 
between  the  two  classes  in  nature  or  probable  causes  of  the 
light  and  velocity  variations. 


CHAPTER  XIII 

LONG  PERIOD  VARIABLES 

PASSING  from  the  study  of  the  short  period  variables  to  those 
of  long  period,  we  find  ourselves  with  an  entirely  different  state 
of  knowledge  both  as  regards  the  light  variation  and  the  speo 
troscopic  evidence.  The  main  facts  will  be  given  briefly  and 
expanded  afterward. 

First,  the  light  variation  is  not  so  regular  as  that  of  the  short 
period  variables.  The  length  of  the  period  is  not  uniform,  the 
magnitude  at  maximum  or  minimum  is  not  the  same  at  differ- 
ent times,  and  the  range  of  variation  is  large,  extending  occa- 
sionally to  five  or  more  magnitudes.  Secondly,  the  spectrum  is 
almost  invariably  of  type  M,  and  is  usually  marked  by  the 
presence  of  bright  hydrogen  lines  at  maximum.  None  of  the 
stars  are  spectroscopic  binaries. 

Very  interesting  data  regarding  the  curves  of  these  stars  can 
be  found  in  Annals,  H.C.O.,1  where  are  published  the  results 
of  observations  of  seventeen  long  period  variables  which  are 
circumpolar  in  this  latitude.  The  following  figures,  which  give 
the  period  of  S  Ursae  Majoris  at  different  times  and  the  accom- 
panying magnitude  at  maximum,  show  irregularities  in  both. 

S  Ursae  Majoris;  Period,    225  days,  Maximum,  7.5  mg. 
206  8.3 

232  7.7 

232  8.0 

218  7.8 

The  variations  in  the  period  are  represented  in  Hartwig  by 
a  sine  term,  showing  that  they  are  periodic  in  character.  His 
elements  are 

J.D.  240  0571  +  226.5  E  +  35  sin  (5°.4  E  +  194°). 
The  magnitudes  of  Mira  Ceti2  at  maximum  are  still  more 
1  Annals,  H.C.O.,  37.  H8.  2  Annals,  H.C.O.,  55, 120-24. 


LONG  PERIOD  VARIABLES  249 

irregular,  as  the  following  list  of  them  taken  at  random  from 
observations  made  during  the  past  forty  years  shows :  — 

1868  5.2  mg.  1886  5.0  mg. 

1869  3.9  1896  4.0 
1875  2.5  1897  3.2 

1879  4.2  1898  2.4 

1885  2.8  1900  3.4 

These  fluctuations,  however,  are  much  greater  than  are  usu- 
ally displayed  by  a  long  period  variable.  The  magnitude  of 
Mira  at  minimum  does  not  vary  so  greatly,  ranging  from  8.5 
to  9.6  in  the  time  mentioned.  Its  period,  like  that  of  S  Ursae 
Majoris,  is  subject  to  large  variations. 

In  the  chapter  on  the  statistical  study  of  variable  star  data, 
many  other  interesting  and  suggestive  facts  will  be  found  in 
regard  to  this  class  of  variables,  and  their  correlations. 

It  has  been  stated  that  very  nearly  all  of  this  class  give  a 
spectrum  of  Secchi's  third  type  with  bright  hydrogen  lines  at 
maximum.  Practically  only  one  star,  Mira,  has  been  investi- 
gated for  motion  in  the  line  of  sight  in  order  to  discover  whether 
it  may  be  a  spectroscopic  binary.  The  evidence  is  entirely 
negative.  Observations  by  Campbell1  made  on  the  dark  lines 
at  the  time  of  the  bright  maximum  in  1897  and  1898  give  a  con- 
stant radial  velocity  and  show  that  the  variable  is  receding 
from  the  sun  at  a  uniform  rate  of  62.3  km.  per  second,  while 
those  made  on  the  bright  hydrogen  lines  at  the  same  time  show 
that  they  have  a  velocity  of  only  about  48  km.  per  second.  This 
would  seem  to  indicate  that  the  envelope  which  is  producing 
the  bright  lines  must  be  moving  toward  the  observer  with  a 
velocity  equal  to  the  difference  in  these  two  rates,  or  15  km. 
per  second.  This  would  confirm  what  might  already  be  imag- 
ined from  the  presence  of  the  bright  lines,  namely,  that  the 
increase  in  brightness  is  due  to  enormous  outbursts  of  hydrogen 
gas  which  occur  with  approximate  regularity. 

The  condition  of  the  star  may  be  somewhat  similar  to  that 
of  the  sun,  only  more  advanced.  It  has  a  tendency  to  form  a 
*  Ap.  J.,  9,  31. 


250          THE  STUDY  OF  VARIABLE  STARS 

heavy  atmosphere,  full  of  clouds  due  to  compounds;  but  just 
as  the  sun-spots  occur  with  seeming  regularity,  and  are  due  to 
a  periodic  instability  in  the  solar  atmosphere,  so  the  balance  of 
forces  in  the  atmospheres  of  the  long  period  variables  is  dis- 
turbed at  certain  intervals,  allowing  outbursts  of  hydrogen  gas 
from  the  interior,  and  resulting  in  a  general  increase  in  the 
brightness  of  the  star.  There  is  some  uncertainty  as  to  what  is 
the  exact  cause  of  the  brightening,  for  it  has  been  stated  that 
simultaneously  with  the  hydrogen  explosions,  there  is  a  general 
brightening  of  the  continuous  spectrum,  showing  that  in  some 
way  the  underlying  photosphere  of  the  star  is  either  brightened 
or  else  shines  through  a  less  dense  layer  of  the  absorbing  atmos- 
phere, since  the  presence  of  the  glowing  hydrogen  is  not  suffi- 
cient to  explain  all  of  the  change  in  the  light  of  the  star.  As  in 
many  other  cases,  determinations  of  radial  velocity  are  much 
to  be  desired,  as  well  as  careful  measurements  of  the  intensity 
of  the  spectrum  at  different  times. 

The  color  of  many  long  period  variables  is  decidedly  reddish. 
The  relation  between  the  color  and  the  length  of  period  will  be 
discussed  in  the  chapter  on  the  statistical  study. 

Pickering  has  separated  a  few  stars  which  are  of  a  peculiar 
character  of  variation,  and  placed  them  in  two  subdivisions  of 
this  group.  He  has  called  them  lib  and  He,  the  first  of  which 
includes  U  Geminorum,  SS  Cygni,  and  SS  Aurigae.  Its  varia- 
tion was  briefly  described  in  Chapter  I.  U  Geminorum  remains 
at  the  minimum  brightness  for  a  large  part  of  the  period,  then 
without  warning  suddenly  rises  to  its  maximum  brightness, 
where  it  remains  for  a  time,  and  then  gradually  fades  away  to 
the  minimum.  The  duration  of  the  maximum  is  not  always  the 
same,  but  the  curve  shows  two  distinct  types  of  maximum,  the 
long  and  the  short.  This  enigmatic  star  has  been  an  object  of 
interest  ever  since  its  discovery  in  1855  by  Hind,  an  English 
observer,  for  the  suddenness  with  which  it  rises  from  the  min- 
imum makes  it  necessary  to  watch  it  constantly.  Two  of 
Hind's  countrymen,  Baxendell  and  Knott,  were  also  interested 
in  this  star,  and  being  in  frequent  communication  with  each 


LONG  PERIOD  VARIABLES 

other,  occasionally  exchanged  telegrams  when  the  star  unex- 
pectedly brightened.  A  very  complete  investigation  of  the  light 
curve  of  this  star  has  recently  been  published  by  J.  Van  der 
Bilt,1  of  the  Observatory  at  Utrecht,  from  which  the  following 
statements  have  been  taken. 

Its  brightness,  which  is  generally  below  13th  magnitude,  sud- 
denly rises  to  about  the  9.5th  magnitude,  remains  above  the 
ordinary  brightness,  that  of  the  minimum,  for  9  or  17  days,  and 
then  repeats  the  process  after  a  period  varying  from  60  to  152 
days.  All  attempts  to  detect  a  law  in  the  changes  of  the  period 
have  failed.  The  maxima  are  of  only  two  types,  the  long  and 
the  short,  and  these  occur  alternately.  The  normal  curves  are 
so  nearly  similar  to  those  of  SS  Cygni  (Chapter  I)  that  they 
need  not  be  repeated  here. 

Since  it  is  so  faint,  even  at  time  of  maximum,  the  spectrum 
has  been  only  imperfectly  observed,  but  is  usually  described  as 
hazy,  and  at  times  as  resembling  Class  F,  the  last  photograph 
having  been  taken  on  February  28,2  1911. 

The  second  star  in  this  group,  SS  Cygni,  has  in  addition  to 
the  long  and  short  maximum  a  third  type,  known  as  anomalous, 
which  is  rather  symmetrical  in  outline.  The  recurrence  of  the 
other  two  types  is  also  remarkable,  for  sometimes  two  long  or 
two  short  maxima  will  occur  in  succession,  and  sometimes  the 
anomalous  form  will  occur  in  their  midst;  but  usually  they  are 
in  the  order  short,  long,  short,  long.  A  very  complete  discussion 
of  this  star  by  Leon  Campbell 3  may  be  found  in  Annals,  H.C.O. 

The  spectrum  is  peculiar  and  is  stated  at  times  to  resemble 
Class  F. 

The  diagram  published  in  Popular  Astronomy  for  April,  1914, 
representing  the  light  variation  for  1913  plotted  from  the  com- 
bined observations  of  members  of  the  American  Association  of 
Variable  Star  Observers,  shows  an  interesting  variation  in  what 
would  ordinarily  be  the  short  maxima,  for  instead  of  following 
the  usual  course  the  upward  slope  is  quite  gradual,  with  a  slight 

1  Recherches  Astronomiques  de  VObservatoire  a"  Utrecht,  in. 

*  Annals,  H.C.O.,  56,  210.  3  Annals,  H.C.O.,  64,  33. 


252         THE  STUDY  OF  VARIABLE  STARS 

irregularity  in  it.  Whether  this  change  is  real  and  permanent 
cannot  be  determined  at  present. 

SS  Aurigae,  the  third  star  of  this  group,  is  quite  faint,  and 
has  not  been  under  observation  long  enough  for  the  collection 
of  many  data  concerning  it. 

The  third  division,  He,  of  the  long  period  variables,  was 
announced  in  H.C.O.,  Circular,  no.  166, 1911,  by  Pickering,  and 
contains  stars  which  are  ordinarily  bright,  but  sometimes  for  a 
year  or  more  become  faint  without  warning,  and  vary  irregu- 
larly until  they  again  attain  their  normal  brightness.  Three 
stars  are  placed  in  this  class  by  him,  R  Coronae  Borealis,  RY 
Sagittarii,  and  SU  Tauri.  Their  spectra  are  also  peculiar  and 
subject  to  changes.  These  three  stars  together  with  the  three  in 
the  preceding  group  merit  careful  attention  on  the  part  of 
variable  star  observers. 

The  irregular  stars  have  usually  a  rather  small  range,  and  are 
somewhat  reddish,  though  varying  from  one  extreme  to  the 
other  of  the  color  scale,  some  being  among  the  reddest  of  the 
stars.  Some  very  bright  stars,  such  as  a  Orionis,  a  Cassiopeiae, 
and  a  Herculis,  are  extremely  capricious  in  their  fluctuations, 
with  a  range  of  about  half  a  magnitude. 

77  Carinae  is  an  irregular  variable  with  a  most  remarkable 
history,  which  is  detailed  at  length  in  Miss  Clerke's  System  of 
the  Stars.  Its  changes,  though  very  great,  take  place  rather 
slowly.  There  is  at  present  nothing  to  indicate  whether  another 
rise  to  extreme  brightness  will  occur  now  or  at  any  time  in  the 
near  future. 

The  peculiarity  of  its  spectrum  causes  it  to  be  classified  with 
temporary  stars  rather  than  with  variables.  A  photograph  of  it, 
taken  at  Arequipa,  Peru,  in  1898,  showed  nearly  all  the  bright 
bands  which  were  in  Nova  Aurigae  in  February,  1892,  with 
about  the  same  intensity.  This  fact  was  later  confirmed  by 
Gill  at  the  Cape  of  Good  Hope.  The  most  recent  observations 
of  its  spectrum  were  made  by  Moore l  in  1913,  at  the  D.  O. 
Mills  station  of  the  Lick  Observatory,  at  Santiago,  Chile. 

1  L.O.B.,  252. 


LONG  PERIOD  VARIABLES  253 

Three  plates  were  exposed,  one  in  1912,  and  two  in  1913.  They 
showed  a  spectrum  of  bright  lines,  and  no  dark  absorption  lines 
could  be  distinguished  with  certainty.  The  identification  of  the 
lines  was  quite  difficult.  Twenty  were  definitely  proven  to  be 
the  enhanced  lines  of  iron.  Titanium  and  chromium  were  also 
recognized,  and  several  other  lines  seemed  to  coincide  with  lines 
in  the  chromosphere.  Several  strong  lines  could  not  be  identi- 
fied, and  on  the  other  hand,  lines  of  helium,  nebular  lines,  and 
Mg.  4481  are  not  present.  Moore  says:  — 

A  comparison  of  the  spectrum  of  t\  Carinae  with  that  of  Novae  in 
the  early  period  of  their  history  indicates  a  close  connection  between 
the  two  spectra.  This  fact,  and  the  great  fluctuation  in  light  exhibited 
by  this  star  in  the  past,ilends  support  to  the  view,  frequently  expressed, 
that  17  Carinae  is  a  Nova.  Further  support  arises  from  the  apparent 
location  of  this  star  in  a  great  nebula. 

The  history  of  the  temporary  stars  has  been  carefully  studied 
and  the  results  presented  in  a  form  well  suited  to  the  general 
reader  by  Miss  Clerke.  Such  stars  need  here  be  considered  only 
in  their  character  of  variable  stars,  the  light  variations  of  which 
are  to  be  studied  like  that  of  any  other  variable.  They  may  be 
described  as  variables  having  but  a  single  maximum.  The  light 
curve  is  characterized  by  a  swift  rise  to  maximum  followed  by  a 
slow  and  irregular  decline  to  minimum.  In  the  case  of  one  star, 
Nova  Persei  no.  2,  there  was  for  some  time  an  unusual  semi- 
regular  fluctuation,  as  can  be  seen  by  an  inspection  of  the  light 
curve  in  Chapter  I.  The  main  interest  centers  perhaps  in  the 
spectrum  of  this  type,  for  the  enormous  displacement  of  the 
lines  indicates  a  velocity  which  seems  almost  unbelievable. 
While  at  first  the  theory  of  two  bodies,  one  emitting  bright  lines 
and  the  other  dark  ones,  and  moving  in  opposite  directions, 
seemed  to  be  accepted,  it  was  with  some  reluctance.  Later, 
that  the  displacements  were  due  to  the  effects  of  pressure  in  the 
atmosphere  or  the  gaseous  envelope  generated  by  the  close 
approach  of  two  masses  of  matter  appeared  more  reasonable. 
At  present,  astronomers  are  waiting  for  more  evidence,  particu- 
larly spectroscopic,  before  advancing  any  further  theory. 


254          THE  STUDY  OF  VARIABLE  STARS 

It  is  also  important  to  get  as  early  a  photograph  of  the  spec- 
trum as  possible,  because  in  the  two  instances  where  this  has 
been  done  before  the  typical  new  star  spectrum  has  developed, 
spectra  of  different  types  have  appeared,  i.e.,  Nova  Persei  no.  2, 
and  Nova  Geminorum  no.  2,  as  mentioned  in  Chapter  I.  The 
former  star  l  showed  first  a  spectrum  of  the  ordinary  Orion 
type,  with  no  trace  of  bright  lines,  except  perhaps  at  the  lower 
edges  of  the  dark  lines.  On  the  next  night  the  K  line  of  calcium 
appeared  to  be  quite  strong,  but  on  the  third  night,  that  is, 
February  24,  the  spectrum  had  completely  changed  into  that  of 
the  ordinary  nova.  A  similar  history  is  recorded  of  Nova  Gem- 
inorum.2 It  was  discovered  by  Enebo  at  Dombaas,  Norway,  on 
March  12,  1912,  and  immediately  announced,  so  that  astron- 
omers were  enabled  to  observe  it  at  once.  Several  plates  of  its 
spectrum  were  taken  at  Harvard,  on  March  13,  at  which  time 
it  was  plainly  of  the  F5  type,  or  midway  between  Procyon  and 
the  Sun.  There  were  some  differences  between  its  spectrum  and 
the  type,  though  very  slight.  No  change  took  place  during  the 
evening.  The  next  evening,  March  14,  the  spectrum  had 
changed,  and  was  transitional  between  that  of  Procyon  and  the 
typical  new  star  type.  The  hydrogen  and  K  lines  had  bright 
broad  bands  on  the  edge  toward  the  red.  On  March  15  the 
transition  to  the  nova  type  was  almost  complete.  These  facts, 
in  connection  with  early  photographs  of  the  region  of  the  sky, 
may  decide  whether  the  star  was  really  a  dark  body,  or  merely 
faint.  Long  exposure  negatives  will  show  the  presence  or  ab- 
sence of  nebulous  matter  about  the  star,  and  throw  more  light 
on  its  origin. 

The  appearance  of  this  preliminary  spectrum  is  the  most  dif- 
ficult fact  to  account  for  in  any  theory  which  has  been  offered  in 
explanation  of  the  new  stars.  Granted  that  the  star  to  which 
it  belongs  was  not  really  a  dark  star,  but  only  extremely  faint, 
it  is  difficult  to  imagine  any  circumstances  which  would  have 
made  it  grow  bright,  as  for  instance,  in  the  case  of  Nova  Persei, 
with  such  great  rapidity,  and  still  retain  the  same  conditions  in 
1  Annals,  H.C.O.,  56,  no.  111.  2  H.C.O.,  Circular,  no.  176. 


LONG  PERIOD  VARIABLES  255 

its  atmosphere  that  existed  before.  Yet  this  is  what  must  have 
happened.  The  causes,  whatever  they  were,  which  generated 
the  great  change,  must  have  greatly  increased  the  heat  and 
light  radiation  of  the  photosphere  of  the  star  before  the  impris- 
oned gases  burst  forth  which  give  its  typical  spectrum.  The 
subject  to  be  investigated,  then,  is  the  exact  point  at  which  the 
tension  of  the  forces  within  can  no  longer  be  held  in  check,  and 
the  disruption  must  take  place.  Only  a  further  study  of  new 
stars  in  their  early  stages  can  give  light  on  this  subject,  and 
since  their  occurrence  is  an  unexpected  phenomenon,  it  is  not 
possible  to  prepare  for  it  except  as  the  astronomer  must  always 
be  prepared  for  the  unexpected. 

The  later  history  of  these  novae  1  is  interesting,  but  rather 
uniform.  The  spectrum  gives  way  ultimately  to  a  nebular  type 
of  bright  bands,  which  grow  faint  rather  rapidly,  leaving  only 
the  continuous  spectrum,  which  is  that  of  an  ordinary  faint 
star. 

The  rest  of  this  chapter  seems  a  suitable  place  in  which  to 
describe  the  collections  of  some  of  the  older  observations  of  long 
period  variables,  which  have  recently  been  edited  and  pub- 
lished. They  are  of  undoubted  value,  for  whatever  may  be  said 
in  regard  to  the  necessity  of  using  the  most  refined  apparatus  hi 
making  observations  of  short  period  variables,  the  Argelander 
method  in  the  hands  of  an  expert  observer  always  furnishes 
material  of  scientific  value  for  long  period  variables.  The  early 
observers  were  among  the  most  skillful  astronomers,  made 
their  observations  carefully,  and  kept  their  records  in  good 
order. 

The  reasons  why  such  series  of  observations  were  not  pre- 
pared for  publication  by  the  observers  themselves  were  usually 
the  same,  lack  of  time  and  means.  The  busy  astronomer  in  an 
observatory  was  ordinarily  occupied  in  more  exact  work  with 
instruments  of  precision,  and  considered  his  observations  on 
variable  stars  as  ranking  only  second  in  value.  There  was  no 
regular  periodical  that  had  room  for  publishing  the  individual 
1  W.  W.  Campbell,  Stellar  Motions,  38. 


256          THE  STUDY  OF  VARIABLE  STARS 

comparisons,  though  results  might  be  welcomed.  This  meant 
that  the  observations  must  be  printed  in  separate  volumes, 
which  could  only  be  done  at  great  expense.  So,  for  one  reason 
or  another,  the  publication  was  not  accomplished  in  the  obser- 
ver's lifetime.  Of  recent  years  much  emphasis  has  been  laid 
upon  publishing  the  original  comparisons,  and  hence  nearly  all 
of  the  longer  series  made  by  the  leading  observers  of  variables 
have  already  been  edited.  Before  describing  the  contents  of  the 
different  volumes,  it  will  be  necessary  first  to  state  what  points 
are  included  in  such  a  work. 

Preparing  for  publication  consists,  first,  in  identifying  all  the 
comparison  stars,  and  obtaining  their  magnitudes,  on  some 
recognized  photometric  scale;  second,  in  examining  the  records 
of  the  original  observations  of  the  stars,  made  night  by  night,  in 
order  to  determine  the  time  of  each,  to  see  that  they  are  cor- 
rectly copied  into  the  ledgers  for  each  star,  and  to  find  the  exact 
method  used  in  making  the  comparisons,  which  was  usually 
some  variation  of  the  Argelander  method;  third,  in  reducing  the 
final  magnitude  or  light  step  of  the  variable  for  each  observa- 
tion, and  in  general  in  studying  all  possible  sources  of  systematic 
error  and  eliminating  them.  The  extent  to  which  these  various 
duties  are  performed  by  the  editor  depends  upon  what  has  al- 
ready been  done  by  the  observer,  and  upon  other  individual 
circumstances.  The  facts  can  be  learned  from  the  editor's 
statement  and  an  inspection  of  the  tables.  The  introduction  to 
each  collection  usually  has  a  concise  statement  of  the  equip- 
ment and  purpose  of  the  observer,  and  relates  how  the  material 
passed  into  the  hands  of  the  editor.  In  the  following  pages  the 
present  writer's  aim  is  to  refer  to  these  points,  to  sketch  briefly 
the  life  of  each  observer,  with  particular  reference  to  his  work 
on  variable  stars,  and  then  to  state  in  what  condition  the  ob- 
servations have  been  published,  so  that  the  investigator  who 
wishes  to  make  use  of  them  will  have  some  idea  of  their  con- 
tents. Such  a  report  is  in  no  sense  a  critical  discussion  of  their 
value.  Among  those  whose  observations  have  been  collected 
and  published  in  this  manner  are  Schonfeld,  Heis,  Krueger, 


LONG  PERIOD  VARIABLES  257 

Schmidt,  Pogson,  Knott,  and  others,  but  first  and  foremost 
stands  Argelander,  with  whom  we  shall  begin. 

Friedrich  Wilhelm  August  Argelander  l  was  born  in  1799,  at 
Memel,  being  the  son  of  a  merchant  whose  family  was  of  Fin- 
nish origin.  The  royal  family  of  Prussia  had  removed  from  Ber- 
lin to  this  part  of  their  kingdom  after  the  events  of  1806,  and 
lived  in  the  house  of  Argelander's  father.  In  the  family  was  the 
crown  prince,  afterwards  King  Frederick  William  IV,  with 
whom  Argelander  formed  a  lasting  friendship.  He  entered  the 
University  of  Konigsberg  and  enrolled  himself  as  a  student  of 
the  science  of  finances.  Very  soon  he  became  more  interested  in 
the  lectures  of  Bessel  than  in  anything  else,  and  begged  to  be 
given  some  computing  in  the  observatory.  His  request  was 
granted,  and  he  became  one  of  BessePs  most  distinguished  pu- 
pils. His  mind  turned  more  and  more  to  questions  of  practical 
astronomy,  and  Bessel  strove  to  strengthen  his  interest  in  the 
science,  and  in  1820  appointed  him  assistant  in  the  observatory. 
It  was  here,  as  described  in  Chapter  II,  that  he  assisted  Bessel 
in  his  experiments  in  making  the  star  chart,  which  gave  him  the 
idea  of  the  great  Durchmusterung.  The  details  of  his  astronomi- 
cal work  need  not  be  presented.  He  was  called  to  the  Observa- 
tory at  Abo  in  1823,  where  he  remained  until  the  town  and  Uni- 
versity buildings  were  destroyed  by  fire.  Later  the  University 
was  removed  to  Helsingfors,  where  he  also  went  as  the  director 
of  the  observatory.  In  1836  he  was  called  to  the  newly  estab- 
lished University  at  Bonn,  where  the  Prussian  Government  had 
decided  to  erect  a  large  astronomical  observatory,  and  its  plan- 
ning and  construction  were  placed  entirely  in  his  hands.  While 
waiting  for  the  plans  to  mature  he  was  obliged  to  content  him- 
self with  a  small  equipment,  and  it  was  to  this  time  of  restricted 
activity  that  we  owe  his  Uranometria  Nova,  and  his  interest  in 
the  study  of  variable  stars.  He  began  in  December,  1838,  with 
an  observation  of  o  Ceti.  He  continued  this  work  with  much 
zeal,  even  while  carrying  on  the  zone  observations  for  the 
Durchmusterung  t  and  imparted  to  the  study  of  variable  stars  a 
i  E.  Schonfeld,  VJS.,  10, 150. 


£58          THE  STUDY  OF  VARIABLE  STARS 

new  dignity  and  value.  Even  at  the  age  of  sixty,  when  he  felt 
that  his  eyesight  was  becoming  feebler,  and  that  his  results  in 
this  line  were  no  longer  valuable,  his  interest  in  the  subject  did 
not  wane.  But  this  was  not  all.  He  early  saw  that  there  was  an 
opportunity  for  interesting  outside  workers  in  the  study  of 
variables,  and  in  1844  published,  in  Schumacher's  Jahrbuch,  an 
article,  "An  Appeal  to  the  Friends  of  Astronomy,"  for  the  pur- 
pose of  urging  them  to  make  interesting  and  useful  observa- 
tions of  the  heavens,  including,  among  other  objects  of  study, 
variable  stars.  A  translation  of  this  portion  of  his  paper,  which 
the  reader  is  urged  to  study,  has  been  published  by  Miss  Can- 
non,1 of  the  Harvard  Observatory.  Argelander  reviews  the 
history  of  variable  stars,  suggests  methods  of  observing  them, 
calls  attention  to  certain  ones  which  are  in  need  of  observation, 
speaks  of  working  with  his  friend  Heis,  and  finally  utters  a  fer- 
vent appeal  to  amateurs  for  co-operation  which  is  worth  quot- 
ing here,  not  only  because  of  its  meaning,  but  because  it  gives  us 
an  idea  of  Argelander  himself,  and  his  enthusiasm  for  his  sub- 
ject:- 

Therefore  do  I  lay  these  hitherto  sorely  neglected  variables  most 
pressingly  on  the  heart  of  all  lovers  of  the  starry  heavens.  May  you 
become  so  grateful  for  the  pleasure  which  has  so  often  rewarded  your 
looking  upward,  which  has  constantly  been  offered  you  anew,  that 
you  will  contribute  your  little  mite  towards  the  more  exact  knowledge 
of  these  stars.  .  .  .  The  observations  may  seem  long  and  difficult  on 
paper,  but  are  in  execution  very  simple,  and  may  be  so  modified  by 
each  one's  individuality  as  to  become  his  own,  and  will  become  so 
bound  up  with  his  own  experiences  that  unconsciously,  as  it  were, 
they  will  soon  be  as  essentials.  As  elsewhere,  so  the  old  saying  holds 
here;  "Well  begun  is  half  done";  and  I  am  thoroughly  convinced  that 
whoever  carries  on  these  observations  for  a  few  weeks  will  find  so  much 
interest  therein  that  he  will  never  cease.  I  have  one  request,  and  it  is 
this;  that  the  observations  shall  be  made  known  each  year.  Observa- 
tions buried  in  the  desk  are  no  observations.  Should  they  be  entrusted 
to  me  for  reduction  or  even  for  publication,  I  will  undertake  it  with 
joy  and  thanks,  and  will  answer  all  questions  with  care,  and  with  the 
greatest  pleasure. 

1  Pop.  Ast.,  20. 


LONG  PERIOD  VARIABLES  259 

So  stimulating  was  his  character  and  so  great  his  influence 
that  it  is  worth  while  to  pause  and  inquire  a  little  more  closely 
into  its  sources.  The  best  appreciation  of  his  life  comes  from 
Schb'nfeld,  his  pupil,  assistant,  and  successor.  It  is  published 
in  the  obituary  notice  given  in  the  Vierteljahrsschrift  and  may 
be  paraphrased  in  the  following  words.  Argelander  was  more 
friendly  to  practical  than  to  theoretical  astronomy.  As  a 
teacher  his  most  interesting  lectures  were  on  the  practical  sub- 
jects, especially  when  he  had  attentive  hearers.  Whenever  he 
had  pupils  in  whose  interest  he  had  confidence,  he  treated  the 
subject  very  penetratingly,  but  he  much  preferred,  at  least  in 
his  Bonn  days  and  before  he  felt  his  years,  to  talk  informally 
with  his  students,  especially  to  those  closest  to  him,  while  walk- 
ing or  when  engaged  in  social  intercourse.  At  such  times  he 
often  went  into  the  most  detailed  and  interesting  explanations, 
for  which  in  a  general  lecture  there  is  neither  time  nor  the  right 
audience.  But  it  was  not  alone  the  great  astronomer  whose 
work  and  teaching  attracted  the  younger  men.  It  was  his  entire 
personality.  It  would  be  impossible  to  depict  this  in  a  few 
words.  Whoever  was  fortunate  enough  to  come  into  contact 
with  him  never  forgot  the  impression  made  by  the  sincerity  of 
his  character,  his  great  kind-heartedness,  his  open,  cheerful 
nature,  and  the  fine  form  of  his  conversation.  Though  familiar 
with  those  of  highest  rank  in  the  kingdom  of  Prussia  from  his 
early  youth,  he  was  nevertheless  a  true  adviser  to  the  least 
beginner,  a  diligent  helper  to  the  pupil,  a  fatherly  friend  to  his 
subordinates,  and  a  cordial  companion  to  his  colleagues.  To 
such  characteristics  may  be  attributed  in  great  measure  his 
success  in  undertakings  requiring  much  co-operation,  such  as 
the  Durchmusterung  catalogue.  Argelander  understood  how  to 
win  the  complete  loyalty  of  his  fellow-workers,  and  to  retain  it 
in  the  work.  He  studied  how  to  remove  every  difficulty  that 
might  become  a  source  of  irritation,  and  never  was  his  own 
activity  greater  than  when  he  noticed,  or  thought  he  noticed,  a 
beginning  of  sluggishness  on  the  part  of  others. 

A  large  portion  of  Argelander's  observations  on  variable  stars 


260          THE  STUDY  OF  VARIABLE  STARS 

he  published  himself  in  various  journals,  the  Astronomische 
Nachrichten  and  others.  Many  appeared  in  the  publications 
of  the  observatory  at  Bonn,  vol.  7,  part  n,  which  covered  a 
period  extending  from  1838  to  1867.  Many  later  observations, 
however,  had  not  been  published  in  his  lifetime,  and  when 
Prof.  Pickering,  during  a  visit  to  the  observatory  in  1883,  dis- 
covered the  fact,  he  asked  the  director,  Schonfeld,  if  he  might 
have  the  privilege  of  copying  them  for  further  discussion.  Per- 
mission was  granted,  the  observations,  about  4000  in  number, 
were  copied,  and  taken  to  Harvard,  where  they  were  reduced, 
and  the  results  published  in  Annals,  vol.  33,  no.  4.  First  comes 
an  extended  study  of  the  comparison  stars,  followed  by  the 
observations  of  sixteen  variables  of  long  period.  The  final  re- 
sults contain  the  Julian  Day,  the  calendar  date,  the  resulting 
magnitudes,  and  the  residuals.  Individual  comparisons  are  not 
given. 

The  observations  of  Schonfeld  were  edited  by  Valentiner,1 
who  succeeded  him  as  director  of  the  observatory  at  Mannheim; 
and  when  later  he  became  director  of  the  Astronomical  Insti- 
tute at  Heidelberg  his  first  work  was  to  publish  the  extensive 
series  of  observations  made  by  Schonfeld.  They  appeared  in 
1900.  Schonfeld,  it  will  be  remembered,  was  Argelander's  assist- 
ant in  making  the  northern  part  of  the  great  Durchmusterung. 
He  was  called  from  Bonn  in  1859  to  become  the  director  of  the 
observatory  at  Mannheim,  in  Baden,  then  just  established. 
The  condition  of  the  country  did  not  permit  the  building  up  of  a 
well-equipped  observatory,  and  he  suffered  from  lack  of  ade- 
quate apparatus  and  assistance.  Nevertheless  he  worked  dili- 
gently in  the  study  of  variable  stars,  and  in  the  course  of  ten 
years  accumulated  a  large  series  of  comparisons.  He  was  then 
called  to  Bonn,  in  1875,  to  become  Argelander's  successor,  and 
at  once  applied  himself  to  the  task  of  preparing  the  southern 
Durchmusterung ',  which  was  the  continuation  of  Argelander's 
catalogue  to  declination  —23°.  This,  in  conjunction  with  his 
labors  as  editor  of  the  Astronomische  Nachrichten,  did  not  allow 
1  Verqff.  d.  Grossh.  Stemwarte  zu  Heidelberg  (Astrometrisches  Institut). 


LONG  PERIOD  VARIABLES  261 

him  time  either  to  continue  his  observations  or  to  prepare  them 
for  publication.  He  fell  ill  not  long  after  the  completion  of  the 
SBD.,  and  died  in  1891.  Valentiner  expresses  the  greatest  ad- 
miration and  reverence  for  him  and  his  accomplishment  amidst 
difficult  surroundings.  "Whoever  remembers  having  seen 
Schonfeld  in  his  little  workroom,  which  scarcely  afforded  space 
for  spreading  out  the  books  which  were  necessary  for  his  work 
of  the  moment;  whoever  recollects  how  unpretentious  he  was, 
and  how  he  could  always  be  contented  with  small  means,  and 
knows  that  Schonfeld  achieved  more  with  his  small  equipment 
than  many  a  richly  endowed  institution  has  accomplished;  to 
him  he  will  be  a  shining  example  of  German  devotion  to  learn- 
ing." He  often  worked  from  early  evening  till  morning  making 
his  observations.  His  accuracy,  as  shown  by  the  error  of  one 
observation,  was  about  .06  mg.  During  the  years  of  his  stay  in 
Mannheim  he  compared  117  variables  with  1100  other  stars, 
and  made  35,963  complete  observations,  with  at  least  5000 
observations  of  the  comparison  stars. 

The  identification  of  the  comparison  stars  presented  the  most 
difficult  task  to  the  editor.  Schonfeld  was  accustomed  to  mark 
with  a  fine  pencil  in  his  Durchmusterung  charts  the  stars  he 
used.  Some  of  the  marks  had  become  partially  erased  in  the 
course  of  time.  Other  drawings  were  on  loose  sheets  which  were 
frequently  misplaced,  so  that  the  identification  of  all  of  the 
comparison  stars  is  not  perfectly  certain.  The  instruments 
employed  were  a  Steinheil  refractor  of  72  Paris  lines,  =  6.4 
inches  aperture,  a  Steinheil  comet  seeker  of  27  Paris  lines, 
=  2.4  inches  aperture,  and  an  opera  glass.  The  estimations 
were  made  in  steps  which  varied  somewhat  with  the  telescope 
employed,  and  on  this  account  he  frequently  used  .5  of  a  step. 
The  publication  is  divided  into  two  parts,  the  first  of  which 
contains  only  the  original  comparisons,  and  not  the  resulting 
magnitudes,  and  the  second  the  identification  of  the  compar- 
ison stars.  In  order  to  make  these  observations  suitable  for 
combination  with  those  made  elsewhere  it  will  be  necessary 
first  to  determine  the  magnitudes  of  the  comparison  stars 


THE  STUDY  OF  VARIABLE  STARS 

according  to  a  standard  photometer  scale.  As  might  be  ex- 
pected, this  work  has  already  been  carried  out  at  the  Harvard 
Observatory,  and  the  results  are  published  in  Annals,  vol.  64, 
no.  3.  The  investigator,  then,  has  only  to  introduce  their 
values  into  the  original  comparisons  and  thus  obtain  the  mag- 
nitudes of  the  variable.  A  series  of  observations  by  Schonfeld, 
of  thirty-two  variables,  had  been  published  during  his  lifetime,1 
but  Pickering  desired  to  re-reduce  them  with  improved  pho- 
tometric determinations  of  the  brightness  of  the  comparison 
stars,  and  they  are  given  in  a  manner  similar  to  that  used  for 
Argelander's  observations,  described  just  previously,  excepting 
that  in  the  table  containing  the  observations  one  additional 
column  is  found,  which  gives  the  light  grade  of  Schonfeld,  as 
published  by  him  in  the  memoirs  just  referred  to. 

Another  astronomer  who  enjoyed  a  friendship  with  Arge- 
lander  was  Eduard  Heis,  the  author  of  the  star  maps  described 
in  Chapter  II.  He  was  a  teacher  of  mathematics  and  physics  in 
the  Gymnasium  at  Cologne,  and  later  at  a  school  in  Aachen. 
Astronomy,  however,  was  his  favorite  science,  and  he  had  for 
his  own  use  a  four-inch  telescope,  but  without  a  dome.  In 
spite  of  his  poor  equipment  his  energy  was  so  great  and  his  eye- 
sight so  clear  that  he  was  able  to  make  a  large  number  of  valu- 
able observations.  Even  in  his  old  age  he  was  accustomed  to 
say  that  he  saw  the  stars  as  sharp  points  without  any  rays.  He 
became  acquainted  with  Argelander  when  the  latter  was  called 
to  Bonn  as  the  director  of  its  observatory,  and  under  his  influ- 
ence he  devoted  much  time  to  ft  Lyrae.  After  his  death  in  1877 
the  manuscript  of  his  observations  was  placed  by  his  family  in 
the  hands  of  Hagen,  who  had  once  been  his  pupil.  In  conse- 
quence of  the  many  questions  which  came  to  him  Hagen  de- 
cided to  edit  and  publish  the  observations.2  He  writes  in  the 
preface  that  the  work,  though  laborious,  was  dear  to  him  be- 
cause it  permitted  him  to  express  his  feeling  of  gratitude  and 

1  Sitzungsberichte  der  Kaiserlichen  Akademie  der  Wissenschaften  (Vienna), 
XLII,  146,  and  XLIV,  503. 
8  Beobachtungen  veranderlicher  Sterne,  by  Eduard  Heis  and  Adalbert  Krueger. 


LONG  PERIOD  VARIABLES 

reverence  toward  his  former  teacher.  In  the  same  volume  are 
included  the  observations  of  Krueger,  which  were  also  en- 
trusted to  him  for  editing,  and  he  writes,  "Their  publication  in 
the  same  volume  with  the  observations  of  Heis,  and  with  the 
same  method  of  reduction  according  to  the  light  steps,  will  be 
a  welcome  gift  to  observers  of  variable  stars,  since  they  were 
begun  and  carried  on  under  the  leadership  of  Argelander,  and 
so  establish  a  tradition  of  the  method  of  this  master." 

Heis's  observations  were  made  largely  with  the  unaided  eye, 
opera  glass,  or  a  small  comet  seeker,  and  his  list  included  sixteen 
stars  known  to  be  variable.  Since  he  was  not  dependent  upon 
an  instrument  for  his  observations  he  frequently  made  compari- 
sons when  on  a  journey,  so  that  we  often  find  references  to  other 
places  than  his  home.  In  1856  he  must  have  gone  on  an  unusu- 
ally extended  trip,  and  taken  as  his  steady  companions  8 
Cephei,  /3  Lyrae,  and  77  Aquilae,  for  we  find  the  names  of  the 
places  at  which  he  stopped  attached  to  the  observations.  As  it 
seems  to  have  been  his  one  long  journey  it  is  interesting  to  dis- 
cover whither  he  went.  On  September  3  he  was  still  in  Minister, 
his  home.  On  September  12  he  was  in  Berlin.  Proceeding  on  his 
journey,  he  made  observations  at  Bodenbach  (between  Dresden 
and  Prague,  at  the  frontier),  Vienna,  and  Kremsmunster, 
reaching  lovely  Gmunden  on  September  30.  From  there  he 
traveled  through  Ischl,  Salzburg,  and  Fraunstein,  reaching 
Miinster  again  on  October  18.  What  charming  memories  must 
have  come  to  him  when  he  looked  over  his  observing  books  and 
came  across  these  entries ! 

In  arranging  the  observations  the  editor  has  placed  first 
under  each  star  the  data  concerning  the  comparison  stars,  their 
names  according  to  Heis,  Flamsteed,  the  ASV.  of  Hagen,  and 
the  BD.\  then  follow  the  steps,  zero  belonging  to  the  first  star; 
the  number  of  times  each  was  used,  and  the  magnitude,  taken 
from  the  PD.9  HP.,  and  other  catalogues  when  possible.  The 
color  is  also  given  when  known.  The  observations  contain  in 
tabular  form  the  calendar  date,  the  Greenwich  M.T.,  the 
comparisons,  the  results  in  light  steps,  and  the  Julian  Days. 


264         THE  STUDY  OP  VARIABLE  STARS 

The  observations  of  Krueger  are  not  so  numerous,  and  are  of 
fewer  stars,  among  them  being  fi  Persei  and  S  Cancri,  which 
were  observed  more  assiduously  than  any  of  the  others.  Krue- 
ger, when  a  young  man,  went  to  the  University  of  Bonn  to  be 
Argelander's  pupil,  and  later  became  his  assistant  and  son-in- 
law.  He  was  made  director  of  the  observatory  at  Helsingfors, 
moved  from  there  to  Gotha,  and  finally  became  editor  of  the 
Astronomische  Nachrichten,  and  director  of  the  observatory  at 
Kiel,  where  he  died  in  1896.  The  editing  of  both  of  these  series 
of  observations  is  done  throughout  with  great  care  and  thor- 
oughness. The  results  are  particularly  useful  as  a  source  of 
illustration  because  the  final  mean  light  step  is  given.  From 
the  work  of  Heis  were  taken  the  observations  which  were  used 
in  Chapters  IX  and  X  for  determining  the  light  scale  of  the 
comparison  stars  and  the  mean  light  curve  of  a  short  period 
variable. 

A  very  large  number  of  observations  of  variable  stars,  which 
extended  over  a  period  of  thirty-five  years,  from  1845  to  1879, 
was  accumulated  by  Julius  Schmidt,1  director  of  the  observa- 
tory at  Athens.  On  account  of  the  favorable  climate  at  Athens 
he  was  able  to  make  observations  with  a  continuity  that  could 
not  be  equaled  elsewhere.  His  entire  collection  was  sent  to 
Vogel,  of  the  Potsdam  Observatory,  after  his  death,  and  a 
copy  was  furnished  at  the  expense  of  the  Harvard  College 
Observatory  to  Pickering,  who  had  them  reduced,  in  part, 
and  published  in  vol.  33  of  the  Annals,  no.  6.  Thirteen  vari- 
able stars  of  long  period  were  observed.  Unfortunately,  not  all 
of  the  comparison  stars  could  be  identified,  but  in  order  not  to 
lose  the  observations  a  method  of  relative  brightness,  as  seen 
by  Schmidt,  was  employed,  rather  than  the  absolute  brightness, 
as  measured  by  the  photometer.  For  this  reason,  and  owing  also 
to  the  fact  that  Schmidt  often  used  large  intervals,  such  as 
seven  or  eight  grades,  in  his  estimations,  only  a  portion  of  the 
observations  was  published,  the  stars  selected  being  four  in 
number.  On  the  other  hand,  the  immense  number  of  the  obser- 

1  A.N.  2577. 


LONG  PERIOD  VARIABLES  265 

vations,  and  the  persistence  with  which  the  comparisons  were 
made,  night  after  night,  and  year  after  year,  give  a  decided 
value  to  the  work.  In  the  table  only  the  Julian  Day  and  the 
final  magnitude  are  published. 

Schmidt  was  born  at  Eutin  in  1825.  While  still  a  schoolboy 
he  showed  a  decided  interest  in  natural  phenomena,  especially 
of  an  astronomical  nature.  His  eyesight  was  keen  and  very 
sensitive  to  all  the  finer  distinctions  of  form,  brightness,  and 
color,  and  he  had  a  special  gift  for  drawing.  He  perceived  that 
his  observations  were  of  scientific  value,  and  devoted  himself  to 
them  with  diligence,  undisturbed  by  the  lack  of  understanding 
displayed  by  his  schoolmates  and  teachers.  In  1846  he  was  ap- 
pointed Argelander's  assistant  at  Bonn,  and  in  1852,  he  became 
director  of  a  private  observatory  in  Olmtitz.  In  1858  he  was 
appointed  director  at  Athens.  One  of  his  finest  pieces  of  work 
was  a  drawing  of  the  chart  of  the  moon,  for  which  task  his 
talents  particularly  fitted  him. 

Harding's  name  is  mentioned  in  this  connection  because  he  is 
credited  with  the  discovery  of  several  of  the  brighter  long 
period  variables,  R  and  S  Serpentis,  R  Aquarii,  R  and  U  Vir- 
ginis,  which  he  made  during  the  years  1811  to  1831.  The  writer 
was  not  able  to  find  any  special  mention  of  his  work  on  varia- 
bles, and  he  is  better  known  for  his  discovery  of  minor  planets, 
for  which  he  made  a  systematic  search,  and  for  his  star  charts, 
which  were  perhaps  the  best  of  his  time.  He  was  a  professor  at 
Gottingen,  where  he  died  in  1834. 

Pogson  is  most  noted,  perhaps,  on  account  of  having  given 
the  value  2.512  to  the  ratio  existing  between  the  brightnesses 
of  two  stars  of  successive  magnitudes;  nevertheless  he  was  a 
steady  observer  of  variable  stars,  and  left  a  large  series  of  obser- 
vations which  has  only  recently  been  published  in  the  Memoirs, 
R.A.S.,  vol.  58.  They  were  prepared  for  publication  by  C.  L. 
Brook,  but  the  volume  contains  an  introductory  note  by  Tur- 
ner, who  had  previously  edited  the  observations  of  Knott  and 
Peek.  Norman  Pogson  was  born  in  1829  in  Nottingham.  He 
was  educated  for  commercial  pursuits,  but  his  natural  scientific 


266          THE  STUDY  OF  VARIABLE  STARS 

interests  led  him  to  study  mathematics  and  astronomy.  He 
finally  took  up  the  work  professionally,  and  held  several  differ- 
ent positions,  one  of  them  being  at  the  Radcliffe  Observatory, 
Oxford,  where  he  became  interested  in  the  study  of  variable 
stars,  and  from  which  place  he  published  his  derivation  of  the 
value  2.512.  Later  he  was  appointed  Government  Observer  at 
Madras,  India,  where  he  died  in  1891.  While  in  India  he  con- 
tinued his  observations  of  variables,  and  at  his  death  the  manu- 
script, containing  4214  observations,  came  into  the  possession 
of  his  nephew,  Mr.  Baxendell,  Jr.  Later  they  passed  into  the 
hands  of  Turner,  who  had  them  copied,  but  was  unable  to  pro- 
ceed with  the  reduction.  Some  time  afterward  a  demand  was 
made  for  early  observations  of  U  Geminorum.  It  occurred  to 
Turner  that  it  might  be  possible  to  secure  the  publication  of  the 
observations  in  the  Monthly  Notices,  one  star  at  a  time,  and  it 
was  with  U  Geminorum  that  he  began.  An  appeal  for  help, 
however,  brought  the  desired  aid  from  Mr.  C.  L.  Brook,  the 
director  of  the  British  Association  of  Variable  Star  Observers, 
which  resulted  hi  their  being  published  in  one  volume. 

Pogson's  observations  are  especially  valuable  because  he  was 
one  of  the  earliest  systematic  observers  of  variables.  His  publi- 
cation contains  observations  and  maps  for  thirty-two  stars, 
thirteen  of  which  he  discovered.  Finding  that  there  was  great 
need  of  charts  for  variables,  he  planned  to  issue  an  extensive 
atlas,  and  made  many  observations  of  comparison  stars  for 
the  purpose.  Unfortunately  the  maps  were  never  published, 
though  a  few  were  printed  for  private  circulation.  Later  ten  of 
them  were  reproduced  under  the  direction  of  Hagen,  and  issued 
as  a  part  of  one  of  the  publications  of  the  Georgetown  College 
Observatory.1  Pogson  also  made  many  color  estimations. 

The  observations  of  Baxendell  (1815-87),  the  brother-in-law 
of  Pogson,  and  friend  of  Knott,  are  in  the  process  of  publica- 
tion, but  not  in  one  volume,  as  with  most  collections.  The 
announcement  of  a  prize  question  on  the  variable  U  Gemino- 
rum, by  the  University  of  Utrecht,  created  a  demand  for  early 
1  Supplementary  Notes  to  the  Atlas  Stellarum  Variabilium,  part  n. 


LONG  PERIOD  VARIABLES  267 

observations  of  this  star,  and  hence  Baxendell's  observations  of 
it  were  prepared  for  publication  by  Turner,  in  whose  hands  the 
entire  set  had  been  placed  by  his  son  for  editing,  and  were  pub- 
lished in  advance  of  his  other  stars.  Turner  reports  that  the 
original  observations  had  not  been  copied  into  ledger  form,  but 
this  task  was  immediately  performed,  and  the  copy  placed  for 
safe  keeping  in  a  different  building.  Baxendell  was  engaged  in 
business  in  Manchester,  but  becoming  interested  in  astronomy, 
he  joined  with  a  friend,  and  together  they  built  an  observatory, 
and  equipped  it  with  a  thirteen-inch  reflector  and  a  five-inch 
refractor.  Later  he  removed  to  Southport,  built  a  new  observa- 
tory for  himself,  containing  a  six-inch  refractor,  and  devoted 
himself  to  observation  of  variables.  He  was  in  constant  com- 
munication with  Pogson,  using  his  star  maps,  and  referring  to 
him  frequently  in  his  notes.  His  earliest  observation  of  U  Gemi- 
norum  was  made  February  1,  1858:  — 

I  believe  the  star  which  I  have  marked  U  will  prove  to  be  the  vari- 
able now  on  its  march  to  another  maximum.  Though  very  small,  it  is 
distinctly  defined,  has  no  haziness  about  it,  and  is  a  dull  yellow  color. 

These  observations  of  Baxendell  were  published  in  the  Monthly 
Notices,  R.A.S.,1  and  in  a  more  recent  number  Turner  pub- 
lishes also  his  observations  of  R  Arietis,2  states  his  reason  for 
continuing  the  work  in  this  form,  and  promises  that  reports  on 
other  long  period  variables  will  follow  from  time  to  time. 

Several  variable  stars  were  discovered  by  Hind  (1823-95), 
though  no  special  series  of  observations  by  him  is  mentioned. 
Born  in  Nottingham,  he  was  early  drawn  to  the  study  of  astron- 
omy. He  held  several  positions  in  observatories,  and  finally 
became  the  superintendent  of  the  Nautical  Almanac  office,  in 
London.  He  was  a  friend  of  Baxendell  and  Knott. 

The  observations  of  George  Knott,  of  twenty-three  long  pe- 
riod variable  stars,  were  edited  by  H.  H.  Turner,  and  published 
in  the  Memoirs,  R.A.S.,  vol.  52,  extending  from  1860  to  1894. 
Mr.  Knott  was  one  of  the  English  scientists  we  so  often  hear  of, 
who,  while  having  independent  means,  are  so  thoroughly  de- 
1  Monthly  Notices,  R.A.S.,  67,  316.  *  Loc.  tit.,  73,  124. 


268          THE  STUDY  OF  VARIABLE  STARS 

voted  to  some  branch  of  science  that,  like  Darwin  or  Huggins, 
they  carry  on  their  line  of  investigation  with  unflagging  devo- 
tion. He  early  became  interested  in  astronomy,  bought  first 
a  four-inch  reflector,  and  later  a  seven-inch  refractor  by  Clark, 
in  1859.  His  records  were  kept  with  scrupulous  neatness,  and  at 
his  death  were  already  partially  reduced,  which  much  facili- 
tated the  work  of  the  editor.  He  gradually  entered  into  an 
extensive  correspondence  with  other  observers  of  variables, 
particularly  the  two  Baxendells,  and  telegrams  were  frequently 
exchanged  between  them,  announcing  unexpected  changes, 
especially  in  such  irregular  variables  as  U  Geminorum.  Some- 
times these  crossed,  as  when  Mr.  Knott  wrote  on  one  occasion 
to  Mr.  Baxendell :  — 

I  was  greatly  amused  at  receiving  your  telegram  this  morning  about 
half  an  hour  after  I  had  started  one  to  you,  and  one  to  Espin,  respect- 
ing our  friend  U  Geminorum. 

After  the  introductory  pages  the  volume  contains  a  list  of  the 
times  of  maxima  and  minima  as  determined  by  Mr.  Knott  and 
entered  in  his  ledgers.  For  each  star  is  given  the  chart  used  in 
observing  it,  which  is  accompanied  by  the  identification  of  the 
comparison  stars  and  their  magnitudes,  and  finally  the  observ- 
ations, which  include  the  Gr.  M.T.  of  the  observation,  the 
light  estimations,  the  resulting  magnitudes,  the  mean  magni- 
tudes, and  remarks. 

Another  volume  of  the  Memoirs  l  contains  observations 
made  under  the  direction  of  Sir  Cuthbert  Peek,  at  his  observa- 
tory at  Rousdon,  near  Lyme  Regis,  during  the  years  1885  to 
1900.  These  also  were  edited  by  Turner,  but  not  until  after  the 
death  of  Peek,  who  had  himself  already  written  the  introduc- 
tion. In  it  he  states  that  the  work  was  in  progress  for  about  ten 
years,  during  which  twenty-two  long  period  variables  were 
under  observation,  and  4133  comparisons  had  been  made. 
These  were  all  done  by  Mr.  Grover,  the  assistant,  though  under 
the  close  personal  supervision  of  the  director.  He  prepared  his 
own  charts,  and  determined  the  magnitudes  of  the  comparison 
1  Memoirs,  R.A.S.,  55. 


LONG  PERIOD  VARIABLES  269 

stars  with  much  care,  revising  them  occasionally.  Argelander's 
method  of  comparison  was  used,  five  stars  being  employed 
whenever  possible,  some  of  which  were  brighter  and  others 
fainter  than  the  variable.  He  found  an  interesting  result  ensu- 
ing when  the  comparison  stars  were  all  either  brighter  or  else 
fainter,  which  may  be  described  in  his  own  words:  — 

Some  of  the  variables  rise  at  maximum  considerably  brighter  than 
any  comparison  star  within  the  same  telescopic  field,  while  others  fall 
at  minimum  below  the  faintest  visible;  thus  it  follows  that  in  the  first 
case  the  comparison  can  only  be  made  with  fainter,  and  in  the  second 
case  with  brighter  stars  than  the  variable  itself.  A  discussion  of  a  large 
number  of  observations  shows  that  when  the  comparison  is  made 
entirely  with  stars  fainter  than  the  variable  the  mean  result  makes  it 
too  bright,  while  when  stars  brighter  than  the  variable  are  employed, 
the  mean  magnitude  comes  out  too  low. 

After  1890  Harvard  charts  and  magnitudes  were  used  for 
some  of  the  stars. 

The  list  of  English  observers  of  variables  may  be  completed 
by  giving  brief  references  to  Pigott  and  Goodricke,  friends  who 
worked  in  the  latter  half  of  the  eighteenth  century.  Goodricke 
was  born  in  Groningen,  Netherlands,  in  1764.  His  father  was 
English,  and  later  returned  to  England  and  settled  in  York. 
In  the  account  of  Goodricke,  written  by  Miss  Clerke,  which  is 
found  in  the  National  Dictionary  of  Biography,  we  find  men- 
tioned several  of  his  articles  on  variable  stars,  which  were  pub- 
lished in  the  Philosophical  Transactions,  but  very  little  is  said  of 
his  life.  A  few  details  of  it  are  given  by  W.  T.  Lynn,  in  a  letter 
to  the  editor  of  the  Observatory.1  He  was  a  deaf-mute,  but  in 
spite  of  his  infirmity  he  received  a  good  education  in  classics 
and  mathematics.  In  a  small  building  in  the  garden  behind  his 
friend  Pigott's  house  the  two  carried  on  together  their  astro- 
nomical observations.  At  eighteen  he  had  discovered  the  period 
and  the  law  of  the  variation  of  Algol,  and  suggested  that  it  was 
due  to  an  eclipse.  He  also  discovered  the  variability  of  ft  Lyrae 
and  S  Cephei,  and  gave  their  periods.  He  died  at  the  early  age 
of  twenty-two. 

1  The  Observatory,  25,  271,  368. 


270          THE  STUDY  OF  VARIABLE  STARS 

Pigott  (flourished  1768-1807)  made  a  variety  of  astronomical 
observations.  He  discovered  the  variability  of  77  Aquilae  in 
1784,  and  found  its  period.  He  also  discovered  the  variability 
of  R  Coronae  and  R  Scuti.  He  published  a  catalogue  of  fifty 
stars,  known  or  suspected  to  be  variable,  in  the  Philosophical 
Transactions.1 

Turning  now  to  the  American  astronomers,  we  find  two  of  the 
early  workers  mentioned  in  this  book  whom  we  wish  to  include, 
though  neither  one  has  left  a  collection  of  observations  to  be 
edited  by  some  devoted  pupil.  They  are  Gould  and  Chandler. 
The  former  is  prominent,  not  only  on  account  of  his  work  on 
magnitude,  but  also  because  he  had  a  great  influence  on  the 
development  of  astronomy  in  America.  Chandler,  on  the  other 
hand,  by  means  of  his  papers  on  various  aspects  of  variable  star 
study,  has  raised  the  theoretical  side  of  the  subject  to  a  digni- 
fied position.  Space  will  be  taken  to  mention  a  few  facts  about 
each.  The  sketch  of  Gould  is  taken  from  an  obituary  notice 
prepared  by  Chandler  for  the  Monthly  Notices.2 

Benjamin  Apthorp  Gould  was  born  in  Boston  in  1824,  and 
graduated  at  Harvard  College.  After  teaching  for  a  year  he 
decided  to  devote  himself  to  a  purely  scientific  career,  and  in 
1845  sailed  for  Europe  to  study  astronomy.  He  was  abroad 
three  years,  during  which  he  spent  three  months  at  the  Green- 
wich Observatory,  four  at  Paris,  a  year  at  Berlin,  another  at 
Gottingen,  four  months  at  Altona,  and  one  at  Gotha.  He  thus 
came  into  contact  as  a  student  with  such  men  as  Gauss,  Encke, 
Wilhelm  Struve,  Hansen,  Peters,  and  Argelander.  As  fellow- 
pupils  he  had  Schonfeld  and  Auwers,  while  Von  Humboldt,  at 
that  time  an  old  man,  became  his  friend.  The  earnest  ambition 
of  this  young  man,  then  just  twenty-one,  must  have  made  a 
great  impression  on  these  European  astronomers,  who  were 
not  so  accustomed  to  the  American  student  as  the  present  day 
German  professor.  The  older  men  grew  interested  in  him  and 
assisted  him  to  obtain  what  he  desired,  and  the  young  ones 
became  his  ardent  friends.  After  his  return  he  maintained  a 
1  Phil.  Trams.,  76, 189.  *  Monthly  Notices,  R.A.S.,  57,  218. 


LONG  PERIOD  VARIABLES  271 

steady  correspondence  with  many  of  the  great  leaders  abroad, 
to  whom  he  was  wont  to  confide  his  projects  and  ambitions,  and 
who  sympathized  with  and  consoled  him  in  return.  Their  inter- 
est was  the  deeper,  not  on  account  of  his  personal  ambitions, 
but  because  of  his  great  desire  to  strengthen  the  position  of 
astronomy,  and  indeed,  of  all  science,  in  America.  With  this  in 
view,  one  of  the  first  projects  he  carried  out  on  his  return  to 
America  was  to  found  the  Astronomical  Journal,  in  1849.  That 
it  really  meant  giving  up  some  of  his  personal  plans  is  shown 
from  a  sentence  in  a  letter  to  Encke :  — 

Though  the  labor  of  supporting  it  will  prevent  me  from  working,  as 
I  otherwise  should,  for  the  advancement  of  my  own  reputation,  still 
the  consciousness  that  I  may  render  now  a  still  greater  service  to 
science  reconciles  me  to  the  abandonment  of  a  great  deal  of  personal 
ambition. 

In  a  letter  to  Von  Humboldt,  written  in  1850,  after  speaking  of 
the  self -distrust  and  intellectual  timidity  in  America,  he  says :  — 

This  I  knew  before  returning  home,  but  realize  it  now  for  the  first 
time  to  its  full  extent.  Therefore  it  is  that  I  dedicate  my  whole 
efforts,  not  to  the  attainment  of  any  reputation  for  myself,  but  to 
serving  to  the  utmost  of  my  ability  the  science  of  my  country. 

He  edited  and  supported  the  Journal,  offering  it  to  astrono- 
mers for  the  publication  exclusively  of  original  investigations. 
It  was  interfered  with,  first  by  the  Civil  War,  in  1861,  and  then 
by  his  expedition  to  South  America,  but  was  revived  again  on 
his  return  in  1885.  As  Chandler  says  of  him:  — 

With  such  universal  and  intimate  connection  with  the  personal 
forces  operating  to  advance  astronomy  in  all  lands,  with  his  intense 
patriotism,  with  his  strong  intellectual  and  moral  traits,  he  could  not 
fail  to  exercise  a  powerful  molding  influence  upon  the  development  of 
American  astronomy. 

Only  a  brief  mention  need  be  made  of  his  astronomical  inves- 
tigations, the  most  important  of  which  were  carried  on  in  the 
Southern  hemisphere.  As  planned  at  first  the  expedition  thither 
was  to  be  provided  for  by  private  subscriptions  from  friends  in 
Boston,  but  through  the  enthusiastic  support  of  its  representa- 


272         THE  STUDY  OF  VARIABLE  STARS 

live  to  the  United  States  the  interest  of  the  government  of  the 
Argentine  Republic  was  aroused,  and  led  to  the  establishment 
of  a  permanent  national  observatory  at  Cordoba.  In  addition 
to  the  Uranometria,  mentioned  in  Chapter  V,  for  which  he 
received  the  gold  medal  of  the  Royal  Astronomical  Society,  he 
carried  on  a  series  of  zone  observations,  covering  the  region 
from  23°  to  80°  south  declination,  prepared  a  general  catalogue 
of  stars,  made  plans  for  a  Durchmusterung,  and  accumulated  a 
series  of  photographic  plates  of  the  principal  clusters  in  the 
southern  heavens,  which  were  taken  to  Cambridge,  where  he 
measured  them  and  prepared  the  results  for  publication. 

The  death  of  Chandler  has  occurred  so  recently  that  no  satis- 
factory estimate  of  his  position  in  American  astronomy  has 
been  published,  and  hence  only  a  few  meager  facts  of  his  life 
can  be  given.  Seth  C.  Chandler  was  born  in  Boston  in  1846.1 
After  graduating  from  the  High  School  he  worked  for  Gould  as 
his  private  assistant,  and  later  entered  the  coast  survey,  in 
1864.  After  having  held  various  business  positions  he  settled  in 
Cambridge  in  1881,  became  associated  with  the  Harvard  Col- 
lege Observatory,  and  resumed  his  astronomical  work.  From 
1886  he  devoted  his  time  entirely  to  investigation.  He  soon  be- 
came interested  in  variable  stars,  their  colors,  and  the  general 
laws  pertaining  to  stellar  variation,  publishing  at  intervals 
catalogues  of  variable  stars  and  other  important  papers.  He  is 
perhaps  best  known  among  astronomers  for  his  discovery  of  the 
variation  of  latitude.  One  of  his  most  important  contributions 
to  astronomical  progress  was  the  editorship  of  the  Astronomical 
Journal,  which  he  took  up  upon  the  death  of  Gould  in  1896.  As 
the  present  editor  has  written,  "If  the  Astronomical  Journal 
was  the  pet  undertaking  of  its  founder,  Dr.  B.  A.  Gould,  it  in  no 
less  measure  became  an  object  of  absorbing  interest  to  Dr. 
Chandler,  when  he  assumed  the  responsibilities  of  the  editor- 
ship, upon  the  death  of  the  founder,  in  1896."  After  1905  ill 
health  overtook  Dr.  Chandler,  and  he  was  not  always  able  to 
perform  his  editorial  duties.  He  died  December  31,  1913. 
1  Ast.  Jour.,  28, 101. 


CHAPTER  XIV 

A  STATISTICAL  STUDY  OF  VARIABLE  STARS 

A  STATISTICAL  study  may  be  defined  as  an  effort  to  correlate 
sets  of  values  in  order  to  discover  if  there  exists  a  dependence  of 
one  set  upon  the  other.  If  such  a  dependence  does  exist  the  fact 
will  be  shown  by  the  trend  of  the  numbers,  or  it  may  be  studied 
by  plotting  the  two  sets  of  values  as  abscissa  and  ordinate  and 
examining  the  resulting  curve.  In  a  study  of  variable  stars  the 
quantities  to  be  correlated  for  each  type  are  number,  length  of 
period,  range,  color,  spectrum,  and  galactic  position.  Among 
the  sources  from  which  the  necessary  data  are  taken  is  the  cata- 
logue contained  in  the  H.C.O.,  Annals,  vol.  55,  published  in 
1907,  which  contains  the  period,  the  magnitudes  at  maximum 
and  minimum,  from  which  the  range  may  be  found,  the  spectral 
type,  and  the  class  of  variation.  Table  II  in  the  same  volume 
contains  the  colors  for  many  variables.  The  Annals,  vol.  56, 
part  vi,  gives  the  spectral  type,  galactic  position,  and  range. 
Hartwig's  Ephemeris  for  1914  gives  the  period  and  range  for  a 
considerably  greater  number  of  stars  than  is  found  in  either  of 
the  Harvard  catalogues.  Hence  it  was  used  as  a  principal 
source  for  some  of  the  statistics.  Long  period  variables  will  be 
discussed  first. 

The  most  obvious  relation  to  be  studied  is  the  distribution 
according  to  the  period.  The  stars  in  Hartwig  were  arranged  in 
groups  according  to  the  length  of  the  period,  the  unit  being 
twenty-five  days.  The  numbers  are  given  in  the  accompanying 
table,  the  first  column  of  which  gives  the  number  of  days  in- 
cluded in  each  group,  and  the  second  the  number  of  stars.  In 
the  third  and  fourth  are  placed  the  mean  range  for  each  group 
and  the  number  of  stars  used  in  forming  this  average. 

It  will  be  noticed  that  the  numbers  in  columns  two  and  four 
are  not  identical;  the  reason  is  that  for  many  stars  no  observa- 


274 


THE  STUDY  OF  VARIABLE  STARS 

TABLE  I 
CLASS  II 


Period 

No. 

Mean  range 

No. 

d      d 

51-  75 

8 

1.1 

8 

76-100 

8 

1.5 

8 

101-125 

10 

1.8 

8 

126-150 

16 

3.3 

15 

151-175 

19 

3.1 

18 

176-200 

23 

4.1 

13 

201-225 

46 

4.5 

35 

226-250 

53 

4.0 

32 

251-275 

51 

4.8 

35 

276-300 

48 

4.8 

30 

301-325 

47 

4.8 

35 

326-350 

49 

4.8 

27 

351-375 

38 

4.3 

27 

376-400 

37 

4.5 

25 

401-425 

21 

5.4 

13 

426-450 

16 

5.3 

10 

451-475 

7 

5.8 

2 

476-500 

9 

4.6 

8 

501-525 

3 

5.4 

2 

526-550 

1 

9.4 

1 

551-575 

3 

7.2 

2 

576-600 

1 

3.2 

1 

601-625 

1 

6.5 

1 

626-650 

0 

— 

0 

651-675 

1 

0.6 

1 

676-700 

1 

1.4 

1 

tions  of  the  real  minimum  have  been  obtained,  but  only  the 
magnitude  below  which  the  minimum  must  lie,  therefore  the 
range  cannot  be  determined  exactly;  e.g.,  U  Cassiopeiae  is 
given  the  magnitudes  of  7.7  and  <C  14.7  at  times  of  maximum 
and  minimum  respectively. 

An  examination  of  the  above  table  shows,  first,  that  the  max- 
imum of  frequency  is  reached  in  the  group  226-250  days, 
though  the  numbers  do  not  vary  much  during  the  entire  inter- 
val 201-350  days,  ranging  from  46  to  53;  second,  that  the  range 
increases  from  1.1  mg.  to  4.5  mg.  during  the  period  51-225 
days,  but  after  that,  is  nearly  constant.  There  are  some  cases  of 
quite  exceptional  range,  but  it  also  happens  that  a  few  stars  of 


STATISTICAL  STUDY  275 

very  long  period  have  a  small  range.  The  exceptions  may  be 
noted. 

Period  Max.  Min.  Range 

V  Delphini  529  da  7.7  mg.  17.1  mg.  9.4  mg. 

SW  Geminorum       698  9.2  10.6  1.4 

ULacertae  659  8.5  9.1  0.6 

The  first  of  these  was  observed  at  minimum  by  Parkhurst,  who 
was  following  it  with  the  forty-inch  telescope  of  the  Yerkes 
Observatory  and  saw  it  reach  a  magnitude  of  17  ±  on  August 
29,  1900.  This  is  the  lowest  magnitude  ever  observed  in  the 
case  of  a  variable.  The  other  two  cases  might  seem  a  little  sus- 
picious, but  the  elements  of  SW  Geminorum  as  published  by 
Hartwig  were  furnished  by  Enebo,  who  is  a  very  reliable 
observer.  They  are  given  in  the  form 

Min  =  2418683  +  698  E. 
No  other  reference  to  the  star  could  be  found  by  the  writer. 

The  next  quantities  to  be  correlated  are  color  and  period.  A 
valuable  article  on  this  subject  has  been  published  by  Beljaw- 
sky,1  who  has  drawn  his  material  from  the  catalogue  in  volume 
55.  From  Table  II  of  that  publication  he  took  the  colors  of 
about  three  hundred  variables,  the  estimations  of  which  had 
been  made  by  several  different  observers,  but  were  based 
largely  upon  the  scales  of  Chandler  and  Osthoff .  The  first  step, 
therefore,  was  to  make  them  homogeneous,  but  this  was  a  mat- 
ter of  some  difficulty,  since  the  stars  common  to  both  scales 
were  few  in  number.  However,  they  were  finally  reduced  to 
Osthoff's  scale,  and  a  table  was  formed  containing  the  stars  in 
order  of  period  and  type  of  variation.  The  first  column  gives 
the  class,  the  second  the  range  of  the  period  for  each  group,  the 
third  the  mean  period  for  each  group,  the  fourth  the  mean 
color,  and  the  fifth  the  number  of  stars  included. 

These  numbers,  when  plotted,  show  a  steady  increase  of  the 
color  with  the  length  of  period.  In  order  to  discover  if  there 
were  systematic  errors  in  this  table  Beljawsky  next  arranged 
the  stars  of  Type  II,  or  the  long  period  variables,  according  to 

1  A.N.  4238. 


276 


THE  STUDY  OF  VARIABLE  STARS 
TABLE  H 


Class 

Period 

M  .  Period 

M  .  Color 

No.  Stars 

V 

Algol  Type 



o!jS3 

13 

IV 

Short  period 

10d 

2.44 

20 

II 

<100d 

80 

3.4 

4 

II 

100<*-200d 

163 

5.11 

24 

II 

200  -250 

226 

4.07 

34 

II 

250  -300 

274 

5.45 

35 

II 

300  -350 

325 

5.83 

49 

II 

350  -400 

374 

6.70 

36 

II 

400  -450 

418 

7.38 

24 

II 

450  -500 

474 

7.9 

7 

III 

Irregular 

— 

7.29 

44 

their  declinations,  but  was  unable  to  find  any  variation  due  to 
this  cause.   He  next  compared  the  color  with  the  maximum 

TABLE  HI 


Limits  of  Mg. 

M.Mff. 

Color 

No. 

>6m 

m 
5.32 

c 
6.8 

11 

6.0-6.4 

6.22 

6.0 

10 

6.5-6.9 

6.66 

6.0 

17 

7.0-7.4 

7.10 

6.10 

27 

7.5-7.9 

7.66 

5.90 

32 

8.0-8.4 

8.12 

5.75 

46 

8.5-8.9 

8.62 

5.56 

33 

9.0-9.4 

9.07 

4.65 

26 

9.5-9.9 

9.5 

3.1 

4 

STATISTICAL  STUDY 


Figure  36 

RELATION  BETWEEN  COLOR  AND  DECREASING  BRIGHTNESS 

brightness  of  the  stars,  and  immediately  there  appeared  a  sys- 
matic  relation,  showing  that  there  was  a  falling  off  in  the  color 
with  the  fainter  stars.  It  will  be  remembered  in  this  connec- 
tion that  Osthoff  stated  that  these  were  all  of  a  uniform  grayish 
color.  The  table  and  the  resulting  curve  are  given  here  because 
of  their  very  great  interest. 

TABLE  IV 


M  .  Period 

M.  Color 

No.  of  Stars 

13d 

c 

2.2 

13 

80 

3.6 

4 

163 

5.04 

22 

226 

4.44 

32 

274 

5.62 

34 

325 

5.79 

47 

374 

6.64 

35 

418 

•»30 

21 

474 

7.8 

7 

Irregular 

7.85 

26 

278          THE  STUDY  OF  VARIABLE  STARS 

On  account  of  this  relationship  the  stars  which  were  brighter 
than  the  fifth  magnitude  and  fainter  than  9.5  mg.  were  excluded 
from  the  original  list,  and  the  curves  of  the  remaining  stars 
were  reduced  to  the  mean  magnitude  8.0.  Table  IV  contains 
the  recomputation. 

Beljawsky  also  investigated  the  relation  between  the  period 
and  range  with  the  same  results  as  those  already  given  above. 
His  values  ended,  however,  at  250  days,  since  the  range  could 
not  be  determined  for  many  stars  of  longer  period  on  account 
of  the  uncertainty  of  the  minimum. 

A  few  exceptional  cases  of  stars  of  long  period  having  color 
low  on  the  scale  are 

Period  Color  Spectrum 

S  Piscium     404  da  1.0  Md  6 

Z  Sagittae    452  2  Md  6 

YDelphini    487  2.0  ? 

Each  of  these  stars  has  the  average  range. 

Another  interesting  correlation  to  make  is  between  spectral 
type  and  color,  or  between  spectral  type  and  length  of  period, 
since  color  and  period  proceed  pari  passu.  Long  period  vari- 
ables nearly  all  have  spectra  of  types  M  or  N,  but  since  not  all 
of  them  are  of  the  same  color,  but  vary  from  yellowish  white  to 
red,  with  a  maximum  at  orange,  it  might  well  be  inquired 
whether  there  exists  any  means  of  subdividing  the  class  into 
groups  which  shall  advance  with  the  color.  A  possible  source  of 
information  on  this  subject  may  be  found  in  Annals,  vol.  56, 
part  vi,  in  the  study  of  stars  having  peculiar  spectra,  which 
was  carried  out  by  Mrs.  Fleming.  She  subdivided  Class  Md 
into  ten  divisions,  but  unfortunately  her  death  occurred  before 
she  had  written  a  complete  description.  However,  the  following 
brief  statement  has  been  published:1 

A  further  examination  of  these  spectra  shows  that  they  can  be 
further  subdivided  into  eleven  groups.  A  classification  was  made 
from  an  examination  of  the  continuous  spectrum,  the  comparative 
brightness  of  the  hydrogen  lines  being  also  carefully  estimated,  always 
assuming  the  brightness  of  HT  as  10.  The  first  class,  of  which  R  Lyncis 

1  Publications  of  the  Astronomical  and  Astrophysical  Society  of  America,  I,  48. 


STATISTICAL  STUDY 


279 


is  the  typical  star,  shows  a  spectrum  resembling  a  Tauri,  and  having 
also  H/3  and  Hy  strong,  bright,  and  nearly  equal,  while  Hs  is  barely 
visible.  The  last  group,  of  which  R  Leonis  is  the  typical  star,  shows  a 
continuous  spectrum.  ...  Of  the  bright  hydrogen  lines  in  R  Leonis 
H/3  is  not  seen,  Hy  is  barely  visible,  and  Hs  is  strongly  marked.  The 
other  classes  form  a  nearly  continuous  sequence.between  these  extremes. 

In  an  effort  to  find  out  if  this  classification  indicated  any 
difference  in  color  the  writer  collected  the  colors  from  volume 
55,  Table  II,  according  to  the  subdivisions  Md  to  Md  10,  and 
took  the  averages,  but  the  results  were  not  especially  satisfac- 
tory, for  there  was  no  marked  increase  among  the  divisions, 
and  indeed  this  was  hardly  expected,  as  the  classification  de- 
pended largely  upon  a  study  of  the  comparative  brightness  of 
the  hydrogen  lines,  and  this  would  not  necessarily  affect  the 
color,  which  would  be  determined  by  the  absorption  in  differ- 
ent parts  of  the  spectrum. 

The  galactic  distribution  of  the  long  period  variables  will  be 
given  in  a  tabulated  form,  together  with  that  of  other  types  of 
stars,  in  a  later  table. 

TABLE  V 
CLASS  IV 


Period 

No. 

Range 

d         d 

d 

0.0-  0.5 

25 

1.03 

0.5-  1.0 

15 

0.91 

1.0-  2.0 

3 

0.67 

2.0-  3.0 

2 

0.75 

3.0-  4.0 

10 

0.78 

4.0-  5.0 

11 

0.82 

5.0-  6.0 

10 

0.93 

6.0-  7.0 

10 

0.80 

7.0-  8.0 

9 

0.81 

8.0-  9.0 

3 

0.67 

9.0-10.0 

5 

0.86 

10.0-15.0 

14 

1.03 

15.0-20.0 

12 

1.48 

20.0-25.0 

8 

1.34 

25.0-30.0 

4 

1.50 

30.0-35.0 

0 

— 

35.0-40.0 

4 

1.62 

40.0-50.0 

4 

1.10 

280 


THE  STUDY  OF  VARIABLE  STARS 


A  study  of  the  short  period  variables  of  Class  IV  may  be 
carried  on  in  the  same  way  as  for  those  of  long  period.  One 
hundred  and  forty-eight  stars  of  this  class  in  Hartwig  were 
divided  somewhat  irregularly  into  groups  in  order  to  show  the 
distribution  and  difference  in  magnitude.  The  mean  range  for 
each  period  was  found  as  before,  but  in  only  one  case  was  it 
necessary  to  omit  a  star  from  the  range  because  the  minimum 
magnitude  was  not  given.  This  was  SX  Persei,  9.1-<C  11.5 
mg.,  Period  4.290.  These  results  will  be  found  in  Table  V  on 
page  279. 

The  Algol  stars  forming  Class  V  were  treated  in  the  same 
way,  and  the  results  are  exhibited  in  Table  VI. 

TABLE  VI 
CLASS   V 


Period 

No. 

Range 

d         d 

d 

0.0-  1.0 

10 

0.98 

1.0-  2.0 

22 

1.12 

2.0-  3.0 

22 

1.63 

3.0-  4.0 

14 

1.99 

4.0-  5.0 

10 

1.51 

5.0-  6.0 

9 

1.51 

6.0-  7.0 

4 

1.08 

7.0-  8.0 

1 

1.0 

8.0-  9.0 

1 

2.7 

9.0-10.0 

2 

1.7  + 

10.0-15.0 

3 

1.20 

15.0-20.0 

1 

1.4 

20.0-25.0 

1 

0.6 

25.0-30.0 

0 

0.0 

30.0-35.0 

3 

1.33 

The  spectral  types  of  all  the  short  period  variables  in  Hart- 
wig  are  given  next,  the  authority  being  volume  56,  no.  vi. 
Following  them  are  certain  exceptional  cases  about  which  no 
special  information  was  furnished  beyond  the  statement  that 
the  spectrum  was  peculiar  for  the  class  of  variation.  It  has  been 
mentioned  that  the  stars  of  Class  II  were  practically  all  of  spec- 
tral type  M  or  N. 


STATISTICAL  STUDY 


281 


Spectral  Types  of  the  Short  Period  Variables 
Class  IV.  Hartwig,  Part  II.  A  11,  Ap  1,  F  18,  F2  2,  F5  13, 

F8  5,  G  23,  G2  4,  G5  8,  K 10,  K5  2,  M?  1,  Mb  2,  N  1.  Total  101. 
Class  V.  Hartwig,  Part  III.  B3  2,  B5  2,  B8  4,  B9  2,  A  60, 

Ap  3,  A3  1,  A5  2,  F  7,  F2  1,  G5  ?.  Total  85. 

Class  IV.   13  Lyrae.   Hartwig,  Part  IV.   B  1,  B2p  1,  B3  1, 

A  7,  Ap  2,  F  2.  Total  14. 

Exceptional  Cases 

Class  IV.  Y  Aurigae,  M?;  ST  Ursae  Majoris,  Mb;  V  Ursae 
Minoris,  Mb;  V  Arietis,  N;  W  Virginis,  cont. 
Class  V.  RT  Lacertae,  G5? 

Class  II.  T  Camelopardis,  Pec.;  SU  Tauri,  G  (this  star  is  in 
Class  He  with  R  Cor.  Bor.);  R  Monocerotis,  ?;  U  Geminorum, 
Pec.,  resembles  Class  F;  R  Sagittae,  G;  SS  Cygni,  Pec.,  resem- 
bles Class  F. 

TABLE  VII 


Spectral  types 

1 

I1 

ft«  * 

i 

o 

1 

0 

1- 

i 

1 

I 

B  and  A 

10 

— 

1 

— 

— 

— 

— 

— 

— 

11 

F 

4(2) 

— 

— 

— 

— 

— 

— 

— 

4 

F  5  and  F  8 

~> 

— 

5(1) 

1 

— 

1 

— 

— 

— 

7 

G 

M  d 

2 

5 

2 

— 

2 

— 

— 

— 

11 

G  5  to  K  5 

° 

— 

— 

3 

2 

— 

— 

— 

— 

5 

K 

— 

— 

2 

— 

2 

— 

— 

— 

4 

Ma 

HH 

M 

— 

— 

1 

1 

1 

2 

— 

— 

5 

Mb 

1 

— 

— 

— 

1 

6 

2 

— 



9 

Me  and  Me  5 

K 

— 

— 

— 

1 

5 

6 

1 

— 

13 

Md 

I 

2 

8 

16 

42 

48 

30 

2 

— 

148 

N 

° 

— 

— 

— 

— 

6 

22 

11 

5 

44 

Total 

10 

8 

19 

25 

47 

71 

62 

14 

5 

261 

282 


THE  STUDY  OF  VARIABLE  STARS 


The  general  relation  between  spectral  type  and  color  for  all 
the  classes  of  variables  has  been  studied  by  Franks,1  who  based 
his  studies  on  the  Harvard  catalogue,  with  additional  informa- 
tion derived  from  other  sources.  He  changed  the  numerical 
scales  of  Chandler  and  Osthoff  into  verbal  terms.  His  results 
may  be  found  in  Table  VII. 

The  galactic  distribution  of  the  variables  was  next  studied 
with  the  aid  of  the  values  given  in  Annals,  vol.  56,  no.  vi.  The 
sphere  was  divided  into  nine  zones  20°  wide,  with  the  central 
one  including  the  Milky  Way.  Table  VIII  contains  in  the  first 
two  columns  the  number  of  each  zone  and  its  limiting  parallels. 
In  the  following  columns  are  to  be  found  the  number  of  vari- 
ables in  each  zone  according  to  their  class,  the  last  column 
giving  the  total  number :  — 

TABLE  VIII 


Zone 

Limits 

// 

IV 

V 

/3  Lyrae 

Novae 

Total 

I 

+  70°  to  +90° 

5 

2 

0 

0 

0 

7 

II 

+50    to  +70 

33 

9 

3 

0 

0 

45 

III 

+30   to  +50 

53 

10 

9 

1 

1 

74 

IV 

+  10   to  +30 

134 

24 

24 

3 

4 

189 

V 

+10    to  -10 

124 

88 

64 

6 

22 

304 

VI 

-10   to  -30 

160 

24 

25 

1 

1 

211 

VII 

-30    to  -50 

79 

9 

7 

1 

0 

96 

VIII 

-50    to  -70 

30 

4 

3 

0 

0 

37 

IX 

-70    to  -90 

14 

0 

0 

0 

0 

14 

In  order  to  determine  the  density  of  the  variables  in  each 
zone  it  is  necessary  to  introduce  the  area  in  square  degrees  of 
each  zone,  which  was  found  in  the  following  manner.  The  area 
of  the  segment  of  a  circle  is 

4?rR2(l  ~  cos  a), 
*  A.N.  4423. 


STATISTICAL  STUDY  283 

where  a  is  the  complement  of  the  arc  which  forms  the  lower 
limit  of  the  segment.  In  this  particular  case  it  is  the  co-latitude 
of  the  base  of  the  zone.  The  ratios  of  the  segments  may  be 
found  by  finding  the  values  of  1  —  cos  a  for  the  desired  lati- 
tudes, and  from  them  the  areas  of  the  zones  simply  by  subtract- 
ing the  area  of  one  segment  from  the  one  adjacent.  The  process 
is  as  follows.  The  zones  in  the  above  list  have  for  their  bases 
galactic  latitudes  70°,  50°,  30°,  10°,  0°.  Their  complements  are 
20°,  40°,  60°,  80°,  90°,  which  are  the  values  of  the  angle  a  to  be 
substituted  in  the  above  formula.  The  resulting  values  are  — 

a          1  —cos  a 

20°         .0603 

40          .2340 

60          .5000 

80          .8264 

90        1.0000 

The  areas  will  then  be  in  the  ratio  of  .0603;  .1737;  .2660;  .3264; 
and  .1736,  or  .3472  if  we  count  the  middle  zone  as  extending 
from  +10°  to  —10°  gal.  lat.  If  we  wish  to  consider  the  central 
zone  as  unity,  which  is  the  most  convenient  method,  they  will 
be  in  the  ratios  .174;  .500;  .766;  .940;  1.000. 

In  order  to  find  the  density  per  zone,  divide  the  number  of 
stars  in  it  by  the  area  and  this  will  give  the  distribution  per  unit 
area.  If  we  divide  the  result  by  304,  the  number  of  stars  in  the 
middle  zone,  the  result  will  be  the  density  compared  with  that 
of  the  Milky  Way.  This  process  is  given  in  Table  IX  for  all 
of  the  variables  taken  together.  The  first  column  gives  the 
number  of  the  zone,  the  second  its  area  relative  to  the  middle 
zone,  the  third  the  number  of  stars  in  each  zone,  the  fourth  the 
number  per  unit  area,  the  fifth  the  density  relative  to  the  mid- 
dle zone  which  contains  the  Milky  Way. 

The  same  kind  of  study  of  the  galactic  distribution  has  been 
made  by  Zinner,1  who  used  1234  stars  found  in  Hartwig's 
Ephemeris  for  1912.  His  values  are  given  in  column  six,  and  in 
the  last  column  are  the  corresponding  values  for  the  distribu- 

i  A.N.  4538. 


284 


THE  STUDY  OP  VARIABLE  STARS 
TABLE  IX 


Zone 

Area 

No. 

stars 

!— 

f  — 

Z 

5 

I 

.174 

7 

40.2 

.13 

.14 

.35 

II 

.500 

45 

90.0 

.30 

.29 

.37 

m 

.766 

74 

96.6 

.32 

.31 

.45 

IV 

.940 

189 

201.6 

.66 

.63 

.68 

V 

1.000 

304 

304.0 

1.00 

1.00 

1.00 

VI 

.940 

211 

224.4 

.74 

.60 

.77 

VII 

.766 

96 

125.3 

.41 

.29 

.47 

vm 

.500 

37 

74.0 

.24 

.20 

.41 

IX 

.174 

14 

80.4 

.26 

.23 

.38 

tion  of  all  the  stars  according  to  Seeliger.  The  comparison  of 
the  values  of  these  last  three  columns  will  show  whether  the 
variable  stars  are  congregated  toward  the  Milky  Way  or  not. 
The  distribution  in  the  three  middle  zones  corresponds  quite 
closely  to  the  distribution  of  all  the  stars,  but  variables  are  not 
plentiful  in  the  regions  about  the  poles.  These  lie  in  the  con- 
stellations of  Cetus  and  Coma  Berenices.  The  former  has  ten 
variables,  a  very  small  number  considering  its  size,  and  the 
latter  only  two.  It,  however,  is  a  rather  small  constellation, 
but  the  adjacent  ones,  Canes  Venatici  and  Bob'tis,  have  eight 
and  fifteen  respectively,  showing  that  variables  are  sparsely 
scattered  about  this  pole  also. 

As  far  as  the  irregular  variables  are  concerned  there  seems  to 
be  very  little  material  that  can  be  tabulated  so  as  to  show  defi- 
nite relations,  though  a  few  general  conclusions  can  be  stated. 
The  range  in  magnitude  is  not  large,  the  average  being  less  than 
2.0  nag.  The  color  is  decidedly  reddish,  lying  on  the  scale  be- 
tween 4.0  and  8.0.  There  seems  to  be  no  special  connection 
between  range  and  color,  but  the  number  of  irregular  variables 
for  which  the  color  has  been  determined  is  rather  too  small  for 


STATISTICAL  STUDY  285 

drawing  definite  conclusions.  A  star  of  range  5.3  has  color  9.1, 
while  a  star  of  range  2.2  has  color  9.7.  There  are  six  variables  in 
this  class  having  color  grade  of  2  or  less,  no  one  of  which  is  de- 
scribed as  being  peculiar.  For  only  two  of  them  is  the  spectral 
type  given.  They  are 

Star  Range  Spectrum     Color 

T  Tauri          9.0  -  12.3          Ma  ?       0 
UMonoc.       5.4-    7.2  K  2 

The  stars  having  largest  range  are 


Star 

R  Cor.  Bor. 
i\  Carinae 
V  Hydrae 
RY  Sagittarii 

Range 
5.8  -  13.8 
-  1    -    7.8 
6.7  -  12.0 
6.6-11.5 

Spectrum 
Pec. 
Nova 
IV 
Pec. 

Color 
2.8 
5 
9.1 
3.5 

The  spectra  of  the  irregular  variables  as  given  in  volume  55 
are  of  Type  M  or  N  with  very  few  exceptions.  Since  there  is 
always  a  possibility  that  a  star  long  called  irregular  like 
u  Herculis,  Sp.  B3A,  may  later  be  found  to  belong  to  the  short 
period  class,  any  marked  peculiarity  in  the  spectrum  should 
at  once  attract  attention  and  suggest  further  observations. 
There  are  eight  of  these,  including  u  Herculis,  just  quoted,  and 
U  Monocerotis,  given  in  the  preceding  list.  They  are:  — 

Star  Range  Spectrum 

X  Tauri  6.6  -    8.1      F  2  G 

S  Doradus       8.2-    9.8      In  cluster  N.G.C.  1910. 

Spectrum  first  type,  having 
H5,  H%  H/3  bright. 

Z  Monoc.         9.0  -  10.1      G  5  K  pec. 
RV  Librae          8.3-    9.0      G? 
p    Cass.  4.4-    5.1      F  8  G 

U  Urs.  Maj.     6.0-    6.5       Pec. 

Among  the  irregular  variables  are  a  few  bright  ones  which 
have  only  a  small  variation. 


Star 

Range 

Spectrum 

Color 

a    Cass. 

2.1  -  2.6 

K 

5.8 

o    Orionis 

0.6-  1.1 

Ma 

7.3 

a    Here. 

3.1  -  3.9 

Mb 

6.0 

R    Lyrae 

4.2-5.1 

Mb 

4.1 

A*    Cephei 

4.0  -  4.8 

Ma 

7.0 

286          THE  STUDY  OF  VARIABLE  STARS 

A  few  statistics  from  Shapley's  investigation  of  eclipsing 
binaries  may  prove  interesting.  The  secondary  minimum  has 
been  observed  in  the  case  of  forty-three  stars  ranging  from  .02 
mg.  to  .8  mg.,  of  which  11  stars  have  a  variation  <C  .1  mg. 

The  figure  of  the  stars  may  next  be  considered.  For  eighteen 
it  is  spherical,  eight  more  show  only  a  slight  flattening,  the  ratio 
between  the  polar  and  equatorial  diameters  being  ^>  .9.  RR 
Centauri  has  the  greatest  flattening,  .631,  while  /3  Lyrae  has 
.758.  In  five  cases  only  is  the  secondary  supposed  to  be  totally 
dark.  The  ratio  of  the  surface  intensities  is  in  most  cases 
]>  1.00.  For  seven  stars  it  is  equal  to  unity,  showing  that  the 
eclipses  are  equal. 

A  few  interesting  miscellaneous  facts  may  now  be  given. 
The  following  list  shows  the  constellations  having  the  largest 
number  of  variables.  Cygnus  is  in  the  lead,  the  last  variable 
given  to  it  in  Hartwig  being  BB,  which  corresponds  to  80.  Next 
in  order  come  Sagittarius,  71;  Scorpio,  65;  Carina,  54;  Andro- 
meda, 43;  Centaurus,  42;  Auriga,  41;  Draco,  39;  Pegasus,  38; 
Hercules,  38;  Aquila,  36;  Orion,  36;  Cassiopeia,  34;  and  Perseus 
34.  These  include  only  the  stars  which  are  lettered  according 
to  the  Argelander  method.  A  few  which  have  other  names  are 
not  included. 

Another  fact  has  to  do  with  the  number  of  variables  discov- 
ered by  a  single  observer.  The  data  for  these  facts  are  taken 
from  volume  55.  Mrs.  Fleming  discovered  108  long  period 
variables;  Anderson,  45;  Madam  Ceraski,  20;  Espin,  19;  Hind, 
18;  Peters,  18;  Williams,  16;  14  were  discovered  at  Bonn,  13  by 
Pogson,  and  11  by  Gould.  Of  Class  IV,  14  were  discovered  by 
Roberts,  10  by  Williams,  7  by  Madam  Ceraski,  and  5  by  Gould. 
Of  Class  V,  13  were  discovered  by  Madam  Ceraski  and  5  by 
Williams.  Of  the  new  stars  8  were  discovered  by  Mrs.  Fleming 
and  2  by  Anderson. 

It  is  interesting  to  note  the  variables  which  were  discovered 
previous  to  the  beginning  of  the  nineteenth  century.  They  are 
arranged  in  order  with  the  name  of  the  discoverer  and  other 
important  facts. 


STATISTICAL  STUDY  287 

1596,  o  Ceti,  Fabricius,  1.7-9.6  mg.,  period  331.6+  da.,  first 
recognized  as  periodic  in  1638. 

1669,  /3  Persei,  Montanari,  2.1-3.2  mg.,  period  2.8+  da., 
discovered  independently  by  Goodricke  in  1782. 

1670,  R  Hydrae,  suspected  by  Montanari,  confirmed  in  1704, 
4.0-9.8  mg.,  period  425.1+  da. 

1686,  x  Cygni,  Kirch,  4.0-13.5  mg.,  period  406.0+  da. 

1782,  R  Leonis,  Koch,  4.6-10.5  mg.,  period  312.8  da. 

1784,  8  Cephei,  Goodricke,  3.7-4.6  mg.,  period  5.3+  da. 

1784,  rj  Aquilae,  Pigott,  3.7-4.5  mg.,  period  7.1+  da. 

1784,  /3  Lyrae,  Goodricke,  3.4—4.1  mg.,  period  12.9+  da. 

1795,  R  Coronae,  Pigott,  5.5  — 12.5  mg.,  period  irregular 
(long). 

1795,  a  Herculis,  W.  Herschel,  3.1—3.9  mg.,  period  irregular. 

1795,  R  Scuti,  Pigott,  4.8—7.8  mg.,  period  irregular. 

These  eleven  stars  may  be  divided  into  practically  two 
classes,  first  those  which  are  of  short  period  and  very  bright  the 
entire  time,  and  second,  those  of  long  period  which  are  very 
bright  at  maximum,  but  telescopic  at  minimum. 

In  conclusion  a  few  general  results  which  seem  to  be  quite 
clearly  defined  may  be  stated  as  follows.  There  is  a  maximum 
of  frequency  for  each  type  of  variable.  For  the  long  period  class 
it  lies  between  200  and  350  days,  for  Class  IV  it  is  less  than  one 
day,  for  Class  V  it  is  from  one  to  four  days.  As  far  as  range  of 
variation  is  concerned  there  is  a  marked  difference  between  long 
and  short  period  variables.  The  latter,  including  Algol,  have  in 
general  a  range  of  less  than  two  magnitudes,  though  there  are 
some  exceptional  cases,  while  for  the  long  period  variables 
the  range  increases  with  the  period  from  51  to  250  days,  and 
then  remains  stationary  at  about  fifth  magnicude,  though  ranges 
of  eight  and  nine  magnitudes  are  known.  There  is  a  marked 
correlation  of  spectral  type  and  class  of  variation.  Long  period 
variables  are  nearly  all  of  Class  M  or  N;  the  stars  of  Class  IV 
range  from  A  to  K,  with  some  exceptions  outside  of  these  limits, 
while  for  Class  V  they  range  from  B  to  G. 

The  property  of  the  variable  stars  least  amenable  to  a  satis- 


288          THE  STUDY  OF  VARIABLE  STARS 

factory  correlation  is  the  color.  This  is  partly  because  the  color 
scales  are  not  uniform,  and  partly  because  the  color  of  the  vari- 
able at  maximum  and  as  it  decreases  are  not  the  same.  Further- 
more, the  relation  between  color  and  spectral  type  does  not 
seem  to  be  as  fixed  as  might  be  expected  from  other  investiga- 
tions. The  long  period  variables,  which  are  all  of  Class  M  or  N, 
vary  hi  color  from  yellowish  orange  to  red,  while  from  their 
spectral  type  they  should  be  quite  red.  The  writer  believes  that 
visual  estimates  of  the  color  are  not  likely  to  give  satisfactory 
results,  but  that  use  must  be  made  of  some  form  of  colorimeter 
or  else  of  the  method  of  Parkhurst  and  Jordan,  described  in 
Chapter  VII,  in  which  the  color  intensity  is  determined  by 
comparing  photographic  and  visual  magnitudes  and  connecting 
these  with  the  spectral  types.  This  might  make  it  possible  to 
arrange  a  classification  of  spectral  type  M  in  which  the  classes 
would  indicate  the  color  with  some  degree  of  certainty. 


CHAPTER  XV 

HINTS  FOR  OBSERVERS 

WHILE  short  period  variables  and  those  with  rapid  changes 
must  be  observed  with  the  best  and  most  refined  devices  for 
measuring  differences  of  brightness,  there  still  remains  a  large 
field  of  opportunity  for  valuable  work  on  the  part  of  an  observer 
with  a  small  telescope,  namely,  the  observation  of  variables  of 
long  period.  Since  these  change  slowly  and  somewhat  irregu- 
larly there  is  no  need  of  the  great  accuracy  with  which  the  vari- 
ables of  the  class  just  mentioned  must  be  observed.  The  num- 
ber of  the  long  period  variables  is  much  greater.  They  cannot, 
except  in  the  case  of  circumpolar  stars,  be  followed  throughout 
an  entire  period  of  variation,  on  account  of  their  being  lost  in 
the  light  of  the  sun  at  certain  times  of  the  year.  Hence  it  is  de- 
sirable that  as  many  astronomers  as  possible  should  share  in 
observing  them. 

Realizing  that  it  is  not  difficult,  with  sufficient  practice,  to 
acquire  skill  in  making  comparisons  by  the  Argelander  method, 
Professor  Pickering,  many  years  ago,  began  issuing  circulars 
inviting  co-operation  in  making  such  observations.  They  con- 
tain directions  for  observing  which  are  very  complete,  and 
cover  practically  all  of  the  difficulties  which  the  observer  is 
likely  to  encounter.  One  of  the  first  pamphlets  was  issued  in 
1891,  and  was  entitled  Variable  Stars  of  Long  Period.  From  it 
the  following  suggestions  and  directions  are  quoted :  — 

A  natural  classification  of  the  variable  stars  seems  to  place  together 
those  having  a  period  of  one  or  two  years.  They  have  many  points  in 
common,  for  instance,  when  near  maximum  the  lines  in  their  spectra 
due  to  hydrogen  are  usually  bright.  This  peculiarity  has  in  several 
cases  led  to  their  discovery,  and  perhaps  furnishes  a  clue  to  the  cause 
of  their  variation  in  light.  Their  color  is  generally  red,  and  the  change 
in  brightness  very  great.  Several  of  them  at  maximum  are  visible  to 
the  naked  eye,  but  at  minimum  become  wholly  invisible,  or  at  least 


290          THE  STUDY  OF  VARIABLE  STARS 

beyond  the  reach  of  any  but  the  largest  telescopes.  This  variation  is 
as  great  as  that  between  the  brightest  and  faintest  stars  visible  to  the 
naked  eye.  Numerous  observations  have  been  made  of  many  of  these 
stars,  but  generally  with  the  object  of  determining  the  times  at  which 
they  attain  their  greatest  brilliancy.  The  rate  of  the  change,  or  form 
of  light  curve,  as  it  is  called,  has  been  comparatively  neglected.  It  is 
the  object  of  the  present  paper  to  provide  a  means  of  supplying  this 
omission. 

Many  astronomers,  provided  in  some  cases  with  excellent  telescopes, 
find  difficulty  in  using  them  in  such  a  way  as  will  really  advance 
astronomical  science.  The  study  of  these  variables  seems  especially 
adapted  to  such  cases.  Except  the  telescope  itself  no  delicate  appara- 
tus like  clock-work  or  micrometer  is  required;  even  divided  circles  are 
not  essential,  although  they  facilitate  observation.  The  variation  in 
brightness  is  also  so  great  that  even  rough  measures  will  have  a  value, 
since  the  laws  regulating  many  of  these  variables  are  almost  entirely 
unknown.  When  the  total  change  in  brightness  is  small,  great  skill  is 
required  to  determine  variations  with  accuracy,  but  less  precision  is 
needed  when  the  variations  amount  to  several  magnitudes,  especially 
as  great  accuracy  seems  to  be  unattainable,  owing  to  the  color  of  these 
stars. 

There  follows  a  description  of  the  Argelander  method,  which 
need  not  be  quoted  here,  since  the  method  has  already  been 
fully  presented  in  an  earlier  chapter.  Some  remarks  on  the 
sources  of  error  to  be  avoided  may  well  be  given :  — 

Evidently  the  comparison  stars  should  be  near  the  variable,  and  a 
very  low  power  should  be  used  so  that  the  apparent  distance  may  be 
small.  Double  stars  and  those  near  brighter  stars  should  not  be  used 
for  comparison,  since  otherwise  errors  will  be  introduced,  whose 
amount  will  vary  with  the  instrument  used.  Since  a  star  near  the  edge 
of  the  field  of  the  telescope  appears  brighter  than  when  near  the  center, 
it  is  better  to  bring  each  star  in  turn  into  the  center  rather  than  to 
place  them  equally  near  the  edge  of  the  field.  When  the  distance  be- 
tween two  stars  is  so  small  that  they  cannot  readily  be  observed  alter- 
nately, as  just  recommended,  it  is  probable  that  owing  to  the  varying 
sensitiveness  of  different  portions  of  the  retina  their  relative  brightness 
will  appear  to  vary  according  to  their  position.  The  head  should 
therefore  always  be  turned  until  the  line  connecting  the  eyes  is  parallel 
to  that  connecting  the  stars,  in  order  that  the  error  may  be  small  in 
all  cases.  Its  amount  may  be  determined  by  selecting  several  pairs  of 
stars,  such  that  in  each  pair  the  stars  shall  be  nearly  equal  in  bright- 


HINTS  FOR  OBSERVERS  291 

ness,  and  one  above  the  other.  Compare  these  stars  with  the  upper 
stars  in  the  successive  pairs,  alternately  to  the  right  and  left,  and 
repeat  with  the  head  turned  the  opposite  way,  so  that  each  pair  is 
measured  once  with  the  upper  star  to  the  right  and  once  to  the  left. 
The  mean  of  the  differences  of  the  results,  when  the  upper  star  is 
turned  to  the  right  and  to  the  left,  will  equal  twice  the  error  due  to 
their  position.  When  the  variable  is  bright  the  comparison  should 
also  be  made  with  the  finder,  with  the  field-glass,  or  with  the  unaided 
eye,  since  it  is  difficult  to  compare  two  very  bright  images. 

After  further  remarks  regarding  the  method  of  finding  the 
region  of  the  variable  and  of  recording  the  observations,  which 
will  be  touched  upon  later,  Pickering  gives  the  positions  of  the 
comparison  stars  for  seventeen  long  period  variables,  which 
are  circumpolar,  and  concludes  with  the  following  recom- 
mendations : — 

Observations  of  these  variable  stars  are  much  to  be  desired,  in  order 
that  the  results  may  be  compared  with  those  obtained  at  Cambridge. 
All  seventeen  may  be  observed  in  two  or  three  hours,  with  proper 
appliances  and  practice.  If  such  observations  could  be  made  several 
times  a  month  by  a  number  of  observers,  we  could  determine  whether 
apparent  sudden  changes  in  light  were  due  to  errors  in  observation  or 
to  actual  variation  in  the  star.  If  observers  with  large  telescopes  would 
undertake  to  follow  these  stars  when  beyond  the  range  of  ordinary  in- 
struments, we  should  obtain  valuable  results  regarding  the  light  at 
minimum.  .  .  .  All  persons  observing  the  variables  or  comparing 
stars  according  to  the  system  described  above,  are  invited  to  send  their 
results  to  Cambridge  for  reduction  and  publication  on  the  same  system 
as  our  own  observations.  It  is  hoped  that  all  can  be  reduced  to  a  uni- 
form scale  of  magnitude,  and  thus  indicate  the  nature  of  the  variation 
of  stars  much  better  than  we  now  know  it.  Should  this  work  commend 
itself  to  astronomers  it  is  hoped  to  extend  it  to  other  variables  of  long 
period. 

The  reader  cannot  help  noticing  how  closely  this  exhortation 
resembles  that  other  written  by  Argelander  nearly  fifty  years 
earlier;  although  the  language  is  less  flowery,  the  interest  and 
the  appeal  are  the  same. 

Ten  years  later,  in  1901,  Pickering  issued  another  circular, 
no.  53,  entitled  Co-operation  in  Observing  Variable  Stars: — 

The  number  of  known  variable  stars  of  long  period  is  now  so  great, 


292          THE  STUDY  OF  VARIABLE  STARS 

and  is  increasing  so  rapidly,  that  the  observation  of  many  of  them  has 
been  greatly  neglected.  Observations  by  Argelander's  method  are  so 
easily  made  that  they  are  specially  adapted  to  observers  who,  for  va- 
rious reasons,  cannot  use  precise  photometric  methods.  In  the  case  of 
variables  of  small  range,  including  those  of  short  period,  and  many 
of  the  Algol  variables,  the  subjective  errors  greatly  diminish  the  value 
of  observations  by  Argelander's  method.  In  these  cases,  also,  the 
periods  and  light  curves  appear  to  be  so  regular  that  continuous 
observations  are  not  needed.  It  appears  to  be  better  to  observe  such 
objects  photometrically  throughout  their  variation,  if  possible,  and 
thus  determine  the  light  curves.  Small  variations  in  the  period  can 
then  be  determined  by  occasional  observations,  at  times  when  the 
light  is  varying  most  rapidly.  Many  of  the  variables  of  long  period 
appear  to  change  irregularly,  and  continuous  observations  are  required 
until  the  nature  of  the  change  is  known.  Moreover,  the  range  is  in 
many  cases  so  great  that  the  errors  of  observation  are  not  sufficient  to 
affect  seriously  the  form  of  the  curve. 

After  describing  the  method  of  making  the  observations,  he 
continues  with  further  remarks  regarding  the  suitable  times  for 
making  the  comparisons:  — 

When  the  variable  is  faint  it  is  impossible  to  observe  it  for  several 
days  every  month  at  the  time  of  full  moon.  At  least  one  observation 
should  be  obtained  in  the  interval  between  the  successive  times  of  full 
moon.  This  can  only  be  done  for  polar  stars,  owing  to  the  proximity 
of  the  sun  at  certain  seasons.  Since  the  periods  of  a  large  portion  of  the 
variables  of  long  period  exceed  half  a  year,  it  is  evident  that  monthly 
observations  will  in  general  give  a  good  idea  of  the  form  of  the  light 
curve.  Of  course  additional  observations  should  also  be  obtained,  but 
failure  to  secure  any  observation  during  a  long  interval  should  be 
avoided  if  possible.  Since  1889  an  attempt  has  been  made  to  observe 
seventeen  circumpolar  variables  north  of  declination  +  50°  at  least 
once  a  month.  These  stars  are  always  above  the  horizon  at  Cambridge, 
so  that  they  can  be  observed  at  all  seasons.  .  .  .  Similar  observations 
have  been  made  of  about  sixty  other  variables,  but  less  regularly.  At 
Arequipa  similar  observations  have  been  made  of  a  large  number  of 
southern  variables.  It  is  much  to  be  desired  that  all  variables  of  long 
period  should  be  observed  in  the  same  way,  or  at  least  so  that  all  can 
be  reduced  to  a  uniform  scale  of  magnitude.  Co-operation  is  necessary 
to  attain  success  in  this  work.  Variables  near  the  ecliptic  can  be 
observed  when  near  the  sun  much  better  at  tropical  stations  than  at 
those  near  the  pole.  The  reverse  is  true  for  polar  variables.  Northern 
variables  can  be  observed  for  a  long  portion  of  the  year  at  northern 


HINTS  FOR  OBSERVERS  293 

observatories,  and  southern  variables  at  southern  observatories.  When 
a  variable  can  be  observed  only  early  in  the  morning  it  is  much  more 
likely  to  escape  observation  than  at  other  seasons.  .  .  .  When  the 
variable  is  bright  it  is  best  observed  with  a  small  telescope,  that  is,  one 
having  an  aperture  of  not  more  than  six  or  eight  inches.  Observations 
of  great  value  could  be  obtained  by  an  observer  with  a  large  telescope 
if  he  was  notified  when  the  star  was  too  faint  to  be  observed  with 
smaller  instruments. 

The  excellent  charts  of  Father  Hagen  are  almost  indispensable  for 
observing  the  stars  when  fainter  than  the  ninth  magnitude.  When  the 
variables  are  bright,  the  need  has  been  felt  here  for  charts  on  a  smaller 
scale,  and  covering  a  larger  region.  After  various  experiments  photo- 
graphic enlargements  have  been  made  of  portions  of  the  admirable 
charts  of  the  Bonn  Durchmusterung.  A  region  three  degrees  square 
surrounding  each  variable  has  been  enlarged  three  times,  thus  giving  a 
map  on  a  standard  scale  of  one  minute  of  arc  to  one  millimetre.  The 
stars  on  these  maps,  while  appearing  coarse  by  daylight,  are  thus  easily 
3een  and  identified  at  night,  without  a  light  bright  enough  to  dazzle 
the  eye.  The  designations  of  the  stars  in  the  sequence  are  marked  on 
these  enlargements,  and  copies  will  be  furnished  at  cost.  Charts  will 
be  furnished  free  of  cost  to  experienced  observers  who  are  ready  to 
co-operate  in  the  above  plan  of  work.  Observations  of  nearly  equal 
value  can  be  obtained  by  those  unaccustomed  to  estimating  intervals 
in  grades.  It  is  only  necessary  to  enter  on  the  charts  the  standard 
magnitudes  of  the  comparison  stars,  and  from  these  to  estimate  di- 
rectly the  magnitude  of  the  variable. 

There  follows  a  list  of  fifty-three  stars,  for  which  the  charts  were 
prepared,  with  the  corresponding  magnitudes  of  their  compari- 
son stars. 

In  Circular  112,  issued  in  1906,  Pickering  again  calls  atten- 
tion to  the  importance  of  continuous  observations  of  variable 
stars  of  long  period,  stating  that  they  are  especially  suitable  for 
observation  by  amateurs  provided  only  with  small  telescopes, 
and  unable  to  devote  much  of  their  time  and  energy  to  astron- 
omy. He  describes  again  the  maps  in  use  at  Harvard,  which  are 
suitable  for  this  work,  the  Durchmusterung  enlargements,  and 
the  Hagen  charts,  and  adds  that  the  latter  "are  supplemented 
by  enlargements  which  have  been  made  of  photographs  of  175 
regions,  taken  with  the  8-inch  Draper  and  Bache  telescope, 
and  show  stars  of  the  12th  or  13th  magnitude.  The  scale  is 


294          THE  STUDY  OF  VARIABLE  STARS, 

20"=. 1  cm.,  and  8X10  prints  on  thick  paper  covering  about  1° 
square  have  been  made  of  them.  They  will  be  sold  at  cost  or 
given  to  observers  qualified  to  use  them." 

He  then  describes  another  method  of  observation  which  was 
substituted  at  the  Harvard  Observatory  for  that  of  Argelander. 
Though  it  has  already  been  mentioned  in  an  earlier  chapter,  we 
may  with  advantage  quote  Pickering's  account  of  it:  — 

A  sequence  of  comparison  stars  is  selected  for  each  variable,  and  the 
photometric  magnitude  determined,  as  described  in  Annals,  37.  This 
magnitude,  to  the  nearest  tenth,  is  entered  on  the  photographic  charts 
described  above.  It  is  well  to  omit  the  decimal  point  to  avoid  mistak- 
ing it  for  a  star.  .  .  .  With  the  chart  thus  marked  the  observation  con- 
sists in  estimating  the  magnitude  directly  by  comparing  it  with  a 
brighter  and  fainter  star.  Thus,  if  found  to  be  distinctly  fainter  than 
a  star  marked  96,  or  of  magnitude  9.6,  and  brighter  than  one  marked 
100,  we  enter  the  magnitude  of  the  variable  9.8;  if  nearly  as  bright  as 
the  brighter  star,  9.7;  and  if  equal  to  it,  9.6.  These  estimates  are  sel- 
dom found  to  differ  by  more  than  a  tenth  of  a  magnitude  from  those 
obtained  by  Argelander's  method.  No  further  reduction  of  these 
observations  is  required,  and  the  light  curve  may  be  constructed  the 
next  day,  using  times  and  magnitudes  as  co-ordinates.  The  observer 
should  never  look  at  the  light  curve  before  making  the  observations, 
as  if  he  knows  what  magnitude  is  to  be  expected  his  observations  will 
have  little  value. 

There  are  about  400  variable  stars  of  long  period  of  the  magnitude 
9.0  or  brighter  at  maximum,  and  having  a  range  of  three  magnitudes 
or  more.  Observations  of  each  of  these  should  be  made  at  least  once  a 
month.  About  300  of  those  north  of  declination  —  30°  are  under 
observation  at  Cambridge,  and  about  40  of  those  south  of  —  30°  at 
Arequipa.  The  pressure  of  other  work  renders  it  difficult  to  follow  all 
of  these  variables  closely,  especially  in  the  case  of  southern  stars. 
Several  observatories  are  now  taking  part  in  this  work,  and  it  is  hoped 
that  the  number  may  be  increased,  especially  for  stars  in  the  southern 
hemisphere.  It  is  only  necessary  that  an  observer  should  be  provided 
with  a  telescope,  preferably  of  4  inches  or  more  in  aperture,  and  be 
able  to  identify  faint  stars  with  certainty.  .  .  .  Careful  watch  of  the 
remarkable  and  often  unexpected  changes  of  a  number  of  stars  is  an 
interesting  occupation,  and  the  fact  that  the  time  is  thus  usefully 
expended  should  induce  many  observers  to  undertake  it  seriously. 

The  reiterated  efforts  of  Pickering  to  arouse  the  interest 
of  astronomers  in  the  observation  of  long  period  variables 


HINTS  FOR  OBSERVERS  295 

finally  met  with  considerable  success,  so  much  so  that  he  was 
able  to  divide  up  the  work  among  different  observers.  In 
1911,  in  Circular  166,  entitled  Co-operation  in  Observing  Vari- 
able Stars,  he  gives  an  account  of  what  had  been  accomplished 
along  this  line,  and  includes  a  list  of  373  variables  of  long  period, 
which  have  a  range  of  at  least  three  magnitudes,  and  are  of  the 
magnitude  9.0  or  brighter  at  maximum.  He  makes  the  follow- 
ing statement:  — 

During  the  years  1906  to  1910  about  17,000  observations  have  been 
made  by  astronomers  connected  with  this  observatory,  of  which  12,000 
have  been  made  by  Mr.  Leon  Campbell.  Six  thousand  observations 
have  been  kindly  communicated  by  other  astronomers,  also  a  very 
large  number  of  observations  have  been  obtained  by  the  members  of 
the  Variable  Star  Section  of  the  British  Astronomical  Association,  and 
by  many  individual  observers.  To  avoid  unnecessary  duplication  and 
to  secure  the  best  results  some  form  of  co-operation  seems  advisable. 
In  the  past  comparatively  few  observations  have  been  secured  of 
the  southern  variables,  and  accordingly  Mr.  Campbell  has  gone  to 
Arequipa  to  undertake  their  observation. 

He  then  mentions  the  names  of  those  who  are  regular  con- 
tributors of  their  observations,  and  earnestly  requests  further 
observations  for  the  following  reason :  — 

While  it  is  often  possible  to  determine  the  form  of  light  curve  from 
observations  made  once  a  month,  much  more  frequent  observations 
are  desired,  especially  in  the  case  of  stars  whose  periods  are  short. 
Accordingly  a  large  number  of  the  variables  have  been  assigned  to  two 
or  more  observers.  Past  experience  has  shown  that  owing  to  clouds, 
moonlight,  and  other  causes,  it  will  be  difficult  even  then  to  avoid 
intervals  exceeding  a  month.  On  the  other  hand,  there  appears  to  be 
an  endless  duplication  in  the  case  of  certain  variables,  and  it  is  hoped 
that  this  may  be  avoided  by  correspondence  between  the  observers. 
Certain  stars  of  special  interest  have  been  assigned  to  all.  This  is  also 
desirable  for  purposes  of  comparison.  It  is  hoped  the  observers  will 
find  that  they  can  follow  many  more  stars  than  those  assigned  to  them, 
as  it  is  not  probable  that  more  observations  will  be  secured  for  a  large 
portion  of  the  stars  than  will  be  needed  to  determine  the  light  curves 
in  each  case.  Observations  of  the  stars  in  the  east,  late  at  night,  or 
early  in  the  morning,  will  be  of  special  value  in  diminishing  the  inter- 
val when  the  star  is  too  near  the  sun  for  observation.  For  similar  rea- 


296          THE  STUDY  OF  VARIABLE  STARS 

sons  observers  having  large  telescopes  could  do  very  useful  work  by 
observing  stars  when  too  faint  to  be  seen  with  small  telescopes. 

For  several  years  the  approximate  magnitudes  of  variables  of  long 
period  have  been  published  each  month  in  Popular  Astronomy.  To 
continue  this  work  it  is  necessary  that  astronomers  should  send  to  the 
Harvard  Observatory,  on  the  first  of  each  month,  a  copy  of  their 
observations,  giving  for  each  star  the  name,  or  designation,  the  date, 
and  the  concluded  magnitude.  Forms  will  be  furnished  for  this  pur- 
pose, and  charts  will  be  given  to  observers  of  the  regions  of  such  vari- 
ables as  they  will  observe  systematically. 

Collections  of  observations  contributed  in  this  way  have  been 
published,  together  with  observations  made  by  the  regular 
staff  at  the  Harvard  Observatory,  and  will  be  found  in  volumes 
of  the  Annals,  37>  57>  and  63. 

From  its  beginning,  in  1893,  Popular  Astronomy  has  served  as 
a  vehicle  of  communication  for  those  who  are  interested  in  vari- 
able star  observations.  The  first  number  contains  the  predicted 
minima  of  eight  variables  of  the  Algol  type,  for  the  months  of 
September  and  October  of  that  year.  These  predictions  were 
continued  through  the  entire  volume,  and  later  other  data 
regarding  variable  stars  began  to  appear.  It  would  be  very 
interesting  to  follow  historically  the  developing  interest  in  this 
work,  as  expressed  by  the  material  contributed  to  this  maga- 
zine, but  space  will  not  permit  it.  The  interest  culminated, 
however,  in  1911,  in  the  formation  of  a  regular  society  for  the 
purpose  of  co-operating  in  the  observation  of  variable  stars, 
which  received  the  title,  "The  American  Association  of  Vari- 
able Star  Observers."  The  first  notice  of  it  comes  in  the  form  of 
some  recommendations  by  the  editor  of  the  journal,  H.  C.  Wil- 
son, called  "What  an  Amateur  Can  Do." 

Many  amateur  observers  would  like  to  do  astronomical  work  of 
scientific  value,  if  they  only  knew  what  they  could  do  with  the  appli- 
ances which  they  have.  Many  spend  the  time  which  they  devote  to 
evening  observation  simply  looking  at  various  portions  of  the  sky, 
exclaiming  to  their  friends  over  the  beauties  of  what  they  see,  and 
perhaps  jotting  down  a  few  notes.  This  is  all  very  well  for  one  who  is 
simply  amusing  himself,  but  it  should  not  be  dignified  by  the  name  of 
"practical  astronomy." 


HINTS  FOR  OBSERVERS  297 

He  then  mentions  several  kinds  of  work  that  the  amateur  can 
do,  the  first  of  which  is  the  study  of  variable  stars.  He  ends 
with  the  query:  — 

Can  we  not  have  in  America  an  association  of  observers  with  a 
" variable  star  section"?  .  .  .  We  invite  correspondence  in  regard  to 
the  matter. 

In  the  next  number  of  Popular  Astronomy  he  states  that  his 
suggestions  seem  to  have  met  with  a  favorable  reception,  and 
the  necessary  correspondence  in  the  direction  of  such  a  section 
was  placed  in  the  hands  of  Mr.  W.  T.  Olcott,  62  Church  Street, 
Norwich,  Connecticut,  who  is  still  the  secretary  of  the  perma- 
nent association.  He  quotes  a  letter  from  Professor  Pickering, 
who  expresses  his  interest  in  the  new  movement,  and  states  that 
while  the  work  done  by  outside  observers  was  being  systemati- 
cally cared  for  at  the  Harvard  Observatory,  still  a  large  amount 
of  useful  work  could  be  done  in  corresponding  with  the  mem- 
bers of  the  association,  providing  for  neglected  stars,  distribut- 
ing photographs,  etc.  "I  believe,"  he  says,  "there  is  useful 
work  for  amateurs  instead  of  merely  looking  at  the  moon  and 
planets."  In  the  succeeding  number  of  the  Popular  Astronomy 
the  new  organization  is  fully  launched,  and  the  first  report  of 
Mr.  Olcott,  the  corresponding  secretary,  appears,  which  con- 
tains suggestions  for  the  future  work  of  the  association :  — 

A  preliminary  publication  should  be  made  of  the  stars  being 
observed  by  each  member  co-operating  in  this  plan.  Thus  each  mem- 
ber will  know  at  the  outset  who  besides  himself  are  observing  certain 
stars  on  his  individual  list,  and  if  he  has  occasion  to  correspond  respect- 
ing certain  observations  he  will  know  at  once  whom  to  address  in  each 
case.  The  further  suggestion  is  made  that  each  member  of  the  associ- 
ation send  in  his  list  to  the  writer  by  the  tenth  of  the  month,  in  order 
that  the  report  may  reach  the  editor  of  Popular  Astronomy  in  time  for 
publication  each  month.  The  list  should  contain,  first,  the  name  and 
address  of  the  observer,  second,  the  type  and  diameter  of  aperture  of 
the  instrument  used,  then  the  name  of  the  star,  the  date  of  observa- 
tion, and  the  estimate  of  its  magnitude.  As  fast  as  lists  are  sent  to 
the  corresponding  secretary  they  will  be  forwarded  to  the  editor  of 
Popular  Astronomy  for  publication,  and  soon  we  hope  to  have  such 
complete  sets  of  observations  that  a  comparison  of  estimates  will  in 


298          THE  STUDY  OF  VARIABLE  STARS 

each  case  be  a  source  of  pleasure  and  profit  to  all  participants  in  this 
plan.  All  observations  made  by  members  of  the  association  will  be 
sent  each  month  to  Professor  Pickering,  Director  of  the  Harvard 
College  Observatory,  who  provides  the  necessary  charts,  and  publishes 
from  time  to  time  discussions  of  the  observations. 

He  then  gives  the  names  of  six  observers  who  have  signified 
their  intention  of  joining  the  association.  In  November,  1911, 
the  first  monthly  report  of  the  new  association  appeared.  By 
this  time  the  membership  had  increased  to  fifteen,  and  in  the 
next  month  to  twenty-five  observers,  representative  of  thirteen 
States  in  the  Union,  and  Canada.  In  a  letter  recently  received 
by  the  author  from  Mr.  Olcott,  the  statement  is  made  that  the 
association  now  has  a  membership  of  thirty-five,  and  that  dur- 
ing the  past  year,  1914,  they  contributed  14,506  observations 
of  255  variables. 

The  British  Astronomical  Association  also  has  a  variable  star 
section,  with  a  membership  of  thirty-nine  observers.  Its  direc- 
tor is  Mr.  Charles  W.  Brook,  of  Meltham,  Yorkshire,  England. 
The  work  of  the  section  for  the  year  1913,  as  published  in  their 
annual  report,  may  be  summed  up  as  follows:  5014  observa- 
tions of  33  long  period  variables  were  made  by  21  observers,  and 
2786  observations  of  6  short  period  variables  by  9  observers. 

Many  useful  hints  may  be  given  to  the  new  recruit  who  is 
taking  up  variable  star  observing,  and  those  who  have  been 
working  at  it  for  a  longer  time  are  only  too  glad  to  contribute 
such  suggestions  as  they  have  found  from  their  own  experience 
to  be  most  useful.  For  convenience  these  may  be  grouped 
under  the  headings  :  (1)  Use  of  Telescope;  (2)  Time;  (3) 
Identification  of  Variable;  (4)  Method  of  Recording;  and 
(5)  Precautions. 

(1)  USE  OF  TELESCOPE.  The  owner  of  a  small  telescope  can 
find  in  any  well-known  handbook,  such  as  that  of  Gibson  or 
Noble,  complete  directions  for  testing  the  lens  of  a  telescope 
and  its  mounting,  and  he  is  supposed  to  understand  the  techni- 
cal part  of  handling  it.  Hence  we  need  here  give  only  such  sug- 
gestions as  apply  directly  to  the  observation  of  variable  stars. 


HINTS  FOR  OBSERVERS  299 

These  include  particularly  the  determination  of  the  width  of 
the  field,  the  penetrating  power,  and  the  method  of  setting.  It 
is  necessary  to  obtain  the  width  of  the  field  in  order  to  know 
how  much  of  a  star  map  is  included  in  the  field  of  view.  This 
may  be  ascertained  as  follows.  Turn  the  telescope  to  a  star 
which  is  near  the  equator  and  not  far  from  the  meridian.  Note 
the  time  required  for  it  to  cross  the  field,  from  one  side  through 
the  center  to  the  other  side.  This  may  be  done  by  using  a  stop- 
watch, or  by  counting  the  beats  of  a  common  clock  which  ticks 
the  seconds.  The  interval,  multiplied  by  fifteen,  to  reduce  it  to 
arc,  will  give  the  width  of  the  field.  This  should  be  determined 
for  the  finder  of  the  telescope  as  well  as  for  each  of  the  powers 
used. 

The  penetrating  power  is  determined  by  finding  the  magni- 
tude of  the  faintest  star  visible  with  the  telescope  under  the 
best  conditions  of  seeing.  This  can  be  accomplished  by  exam- 
ining on  a  clear  and  steady  dark  night  a  field  of  stars,  the  mag- 
nitudes of  which  have  been  well  determined  photometrically. 
One  such  field  is  the  Pleiades,  a  map  of  which  can  be  found  in 
Ball's  Popular  Guide  to  the  Heavens.  Their  magnitudes  and 
positions  are  given  in  an  excellent  article  by  Muller  and  Kempf , 
entitled  "The  Brightness  of  96  Stars  in  the  Pleiades,"  in  A.N. 
3587-88. 

This  group  of  stars  has  been  investigated  more  frequently 
than  any  other  in  the  sky,  but  there  are  other  fields  which  can 
equally  well  be  used  for  this  same  purpose.  Some  of  the  maps 
which  have  been  prepared  particularly  for  variable  stars  will 
serve  this  purpose,  if  we  can  be  sure  that  the  magnitudes  of  the 
faint  stars  have  been  well  determined.  Parkhurst,  in  his  Re- 
searches in  Stellar  Photometry,  has  investigated  with  a  wedge 
photometer  attached  to  the  forty-inch  telescope,  the  magni- 
tudes of  the  faint  comparison  stars  of  twelve  variables,  maps  of 
which  may  be  found  in  that  publication.  These  furnish,  per- 
haps, the  best  determinations  of  magnitudes  of  the  faint  stars. 

In  locating  the  variable,  if  the  telescope  has  no  circles  it 
should  be  pointed  directly  at  the  region  in  the  sky  where  the 


300          THE  STUDY  OF  VARIABLE  STARS 

variable  is  supposed  to  be.  In  case  it  has  circles  the  setting 
should  be  made  in  the  following  manner.  From  the  right  ascen- 
sion of  the  variable  and  the  sidereal  time  find  the  hour  angle 
from  the  equation,  — 

Hour  angle  =  Sidereal  time  —  Right  ascension. 
If  the  sidereal  time  is  greater  than  the  right  ascension,  the  star 
has  already  crossed  the  meridian,  and  the  hour  angle  is  west,  or 
positive.  If  the  sidereal  time  is  less  than  the  right  ascension,  the 
star  has  not  yet  come  to  the  meridian,  and  the  hour  angle  is  east, 
or  negative.  In  order  to  perform  the  substraction  indicated  in 
the  second  member  of  the  above  equation,  it  is  sometimes  nec- 
essary to  add  twenty-four  hours  to  the  sidereal  time,  but  in  any 
case  the  resulting  hour  angle  must  be  less  than  twelve  hours. 

If  the  telescope  have  no  clamp  in  declination,  it  will  be  con- 
venient, after  setting  in  declination,  to  set  in  hour  angle  by  tak- 
ing hold  of  the  end  of  the  declination  axis,  thus  avoiding  touch- 
ing the  telescope  itself  and  disturbing  the  setting  in  decimation 
already  made.  The  order  for  the  observer  to  follow  in  setting 
on  a  variable,  then,  would  be :  first,  find  the  hour  angle,  decide 
whether  the  star  is  east  or  west  of  the  meridian,  and  place  the 
telescope  in  the  proper  position  with  reference  to  the  pier. 
Next  set  in  declination,  and  then  set  in  hour  angle.  If  the  vari- 
able is  not  in  the  field  of  the  main  telescope,  it  should,  if  the 
setting  is  correct,  at  least  be  near  the  center  of  the  field  of  the 
finder.  In  handling  the  telescope,  the  observer  should  accustom 
himself  to  swinging  it,  either  in  right  ascension  or  declination, 
so  that  he  can  move  it  in  either  co-ordinate  without  disturbing 
the  other.  This  is  facilitated  if  the  observer  holds  his  eyes 
parallel  to  the  diurnal  motion  of  the  stars,  for  then  the  motion 
in  right  ascension  will  be  parallel  to  that  direction,  and  the 
motion  in  declination  perpendicular  to  it. 

(2)  TIME.  Usually  the  amateur  observer  has  no  sidereal 
clock  to  use  in  determining  the  hour  angle.  The  sidereal  time 
must  then  be  obtained  by  reducing  the  mean  solar  time  to  the 
corresponding  sidereal  time,  with  the  aid  of  the  American 
Ephemeris.  The  simplest  way  to  proceed  is  to  select  some  in- 


HINTS  FOR  OBSERVERS  301 

slant  of  time  early  in  the  evening,  such  as  7  P.M.,  and  change 
this  into  the  corresponding  sidereal  time.  Set  the  ordinary 
clock  by  the  result  and  it  will  serve  during  the  hours  of  observa- 
tion. Since  common  clocks  read  only  to  twelve  hours,  and  not 
to  twenty-four,  when  the  sidereal  time  is  greater  than  twelve 
the  observer  must  remember  to  add  twelve  to  the  face  time  of 
the  clock.  The  Ephemeris  for  1916,  which  is  already  in  print, 
gives  (on  pp.  713-14)  the  method  for  computing  the  change  of 
time  with  an  example.  As  this  can  easily  be  procured  from  the 
Government  Printing  Office,  Washington,  no  further  explana- 
tion need  be  given  here.  The  observer  must  know  approxi- 
mately his  longitude.  Care  must  be  taken  that  the  mean  time 
clock  is  set  according  to  U.S.  Observatory  time,  which  can  be 
obtained  from  any  reliable  jeweler. 

(3)  IDENTIFICATION  OF  VARIABLE.  The  first  step  to  be  taken 
in  the  identification  of  variable  star  fields  is  to  learn  how  to  hold 
the  map  properly.  Various  observers  have  different  ways  of  do- 
ing this.  The  rule  which  is  given  here  is  one  which  the  writer 
has  used  for  many  years,  and  considers  the  simplest  and  most 
direct  way  of  accomplishing  the  desired  purpose.  It  is  as  fol- 
lows. Every  star  map  is  made  for  use  with  the  inverting  tele- 
scope, according  to  which  the  lower  edge  of  the  diagram  is 
north,  the  right  hand  is  east,  the  upper  edge  south,  and  the  left 
edge  west.  The  eastern  part  of  the  field  is  also  known  as  the 
following  edge,  and  the  western  as  the  preceding.  It  will  be  noted 
that  the  letters  N,  E,  S,  W  follow  each  other  in  an  anti-clock- 
wise order.  It  should  also  be  remembered  that  by  diurnal  mo- 
tion the  stars  move  across  the  field  in  a  direct  line  from  east  to 
west.  Furthermore,  in  different  parts  of  the  sky  the  inclination 
of  this  east  and  west  line  to  the  horizon  will  vary.  Since  in 
looking  in  the  telescope,  the  observer  generally  places  his  eyes 
parallel  to  the  horizon,  this  line  will  make  an  angle  with  the 
line  passing  through  the  two  eyes,  and  the  map  must  be  turned 
so  as  to  allow  for  it.  Therefore,  on  first  looking  into  the  tele- 
scope, the  observer  should  allow  the  stars  to  travel  across  the 
field  by  their  diurnal  motion.  Having  noted  the  direction  of 


302         THE  STUDY  OF  VARIABLE  STARS 

this  line,  he  should  hold  the  map  so  that  the  east-west  line  shall 
be  parallel  to  the  east-west  line  in  the  telescope.  West  can  be 
very  easily  distinguished  from  east  because  the  stars  come  in  on 
the  eastern  edge  of  the  field,  and  pass  out  at  the  western  edge. 
When  west  is  once  located,  the  other  directions,  north,  east,  and 
south,  follow  each  other  in  the  anti-clockwise  direction.  This 
rule  seems  to  be  a  very  easy  one  to  remember,  because  it  is  in- 
variable, no  matter  in  what  part  of  the  sky  the  observer  is  look- 
ing, and  hence  the  directions  east  and  west  and  right  hand  and 
left  hand  need  not  be  considered. 

The  writer  has  also  found  it  very  advantageous,  in  comparing 
the  telescopic  field  with  the  star  map,  to  incline  the  head  at  such 
an  angle  that  the  line  joining  the  eyes  is  parallel  to  the  east- 
west  line.  If  the  telescope  is  provided  with  circles,  and  the 
diameter  of  the  field  of  view  has  been  obtained,  it  is  a  compara- 
tively simple  matter  to  identify  the  variable,  for  when  once  the 
telescope  is  set  for  the  right  position  the  variable  should  be  in 
the  center  of  the  field,  and  can  be  recognized  by  its  position 
with  reference  to  some  conspicuous  group  of  stars  in  the  field. 
It  naturally  happens  that  some  fields  are  more  easily  recognized 
than  others,  but  after  some  experience  the  observer  should  have 
no  difficulty  in  being  sure  of  the  identification  of  the  variable. 
If  the  telescope  is  not  provided  with  circles  that  matter  is  not  so 
simple. 

Since  the  writer  has  always  been  fortunate  enough  to  observe 
with  a  mounted  telescope  she  has  had  no  experience  in  pointing 
directly  at  a  variable,  and  has  made  use  of  the  following  hints, 
which  were  kindly  furnished  her  by  Mr.  Olcott:  — 

The  first  step,  if  the  glass  is  not  furnished  with  circles,  is  to  plot  on 
the  star  atlas  (Klein,  Schurig,  or  Upton)  the  position  of  the  variable. 
Make  a  tracing  showing  its  exact  position  as  regards  lucid  stars  near 
it  in  the  sky.  Use  the  lowest  power  ocular  (a  power  of  thirty  on  a 
three-inch  glass  is  excellent).  Direct  the  telescope  at  the  approximate 
location  of  the  variable,  and  slowly  sweep  in  this  region  until  you 
locate  the  exact  field  covered  by  the  chart.  Identification  of  the  field 
is  greatly  facilitated  by  connecting  with  lines  on  the  chart  the  brightest 
stars  on  the  field.  This  yields  several  geometrical  figures,  and  with  a 


HINTS  FOR  OBSERVERS  303 

mental  picture  of  these  in  sweeping  with  the  telescope  there  should  be 
little  difficulty  in  locating  the  variable. 

In  general  it  may  be  said  that  the  observer  should  make  use  of 
some  star  which  can  easily  be  identified  with  the  naked  eye. 
This  should  be  placed  in  the  finder,  and  then,  by  alignment,  or 
passing  from  group  to  group,  the  field  immediately  surrounding 
the  variable  can  be  found. 

(4)  METHOD  OF  RECORDING.  The  entries  made  in  the  record 
vary  more  or  less  with  the  circumstances  of  the  observer.  They 
should  include  the  date,  the  instrument  and  power  used,  the 
condition  of  the  sky,  whether  the  seeing  is  good,  mediocre,  or 
poor,  whether  there  is  bright  moonlight,  whether  it  is  misty  or 
frosty,  whether  the  star  images  are  sharply  defined,  or  poor  and 
fuzzy;  the  name  of  the  star,  and  the  comparisons  (here  the  char- 
acter of  the  entry  will  depend  upon  the  method  of  observation). 
The  time  of  the  observation  should  next  be  given.   In  the  case 
of  long  period  variables  this  need  be  stated  only  to  the  nearest 
tenth  of  a  day;  but  with  variables  the  period  of  which  is  thirty 
days  or  less,  including  the  very  short  period  variables,  the  near- 
est minute  should  be  given  also.  If  the  circles  are  used  for  set- 
ting it  is  often  convenient  to  record  the  sidereal  time  and  the 
hour  angle  of  the  observation. 

(5)  PRECAUTIONS.    Several  precautions  have  already  been 
mentioned  in  Chapter  VI,  and  also  appear  in  the  Circulars 
issued  by  Professor  Pickering.  The  following  are  given  by  Park- 
hurst  as  essentials  for  good  visual  comparisons.  Since  some  of 
them  merely  repeat  what  has  been  said  before  quite  fully,  they 
need  only  be  briefly  mentioned  here.   First,  the  line  joining  the 
two  stars  to  be  compared  should  be  parallel  to  the  line  of  the 
eyes.  Second,  two  or  three  comparison  stars  should  be  used  at 
each  observation,  if  they  can  be  found  in  proper  distances  and 
magnitudes.  Third,  the  stars  to  be  compared  should  be  in  the 
same  field.   Fourth,  the  interval  in  brightness  should  be  less 
than  half  a  magnitude.  If  this  limit  is  exceeded,  the  compari- 
sons should  be  weighted  in  the  reduction  inversely  as  the  inter- 
val. Fifth,  prejudice  which  might  arise  from  anticipating  the 


304         THE  STUDY  OF  VARIABLE  STARS 

star's  expected  change  should  be  avoided  by  postponing  reduc- 
tions till  the  maximum  or  minimum  is  complete.  Sixth,  the 
comparison  of  two  bright  stars  should  be  avoided  by  reducing 
the  aperture  by  a  suitable  cap  (this  may  sometimes  be  accom- 
plished by  using  the  finder).  Seventh,  light  in  the  eyes  should 
be  avoided  by  using  for  recording  a  one-candle-power  incandes- 
cent lamp,  so  shielded  as  to  illuminate  faintly  a  circle  one  or 
two  inches  in  diameter  on  the  record  book.  In  another  place 
Parkhurst  suggests  using  a  red  light.  We  have  already  stated 
that  in  the  case  of  colored  variables  it  is  essential  to  take  a  long 
and  steady  look  at  the  star,  since  it  grows  brighter  by  so  doing. 

Two  miscellaneous  precautions  may  be  added  here.  One  is 
intended  for  the  observer  who  is  using  Hagen's  charts.  The 
scale  for  Series  IV  is  different  from  that  of  Series  I-III,  the 
former  being  one  half  that  of  the  latter,  that  is,  the  field  of  view 
of  the  telescope  covers  a  circle  of  half  the  diameter  for  Series 
IV,  since  the  side  of  the  square  is  60'  instead  of  30'.  This  may 
also  be  described  by  saying  that  the  stars  on  the  maps  of  Series 
IV  will  be  further  apart  in  the  field  of  the  telescope,  apparently, 
than  they  are  for  the  maps  in  Series  I-III. 

The  second  comes  from  Yendell,  who  states  that  in  combin- 
ing different  comparisons  of  a  variable  made  on  the  same  night, 
he  usually  assigns  weights  depending  upon  the  step  intervals  of 
the  individual  comparisons,  as  follows: l 

It  is  considered  that  the  comparisons  most  likely  to  be  carefully 
studied  and  decided  upon  are  those  showing  the  equality  of  the  vari- 
able with  its  comparison  star,  so  that  to  such  comparisons  should  be 
assigned  the  highest  weight.  For  each  step  of  interval  this  weight  is 
supposed  to  be  less.  From  the  nature  of  the  case  any  such  scheme  of 
weights  is  an  arbitrary  one.  The  writer  has  used  for  many  years  a 
system  of  weights  suggested  to  him  by  a  very  experienced  observer. 
The  weight  of  a  comparison  showing  equality  is  assumed  as  8,  so  that 
when  the  interval  shall  be  as  much  as  four  or  five  steps  the  weight  shall 
still  be  sufficient  to  give  it  its  fair  share  of  influence  in  forming  the 
mean.  For  each  step  of  interval  the  weight  is  reduced  by  one.  The 
difference  between  the  result  obtained  by  this  method  and  by  the 
ordinary  method  of  taking  the  simple  mean  is  not  sufficient  to  be 
1  Pop.  Ast..  14,  540. 


HINTS  FOR  OBSERVERS  305 

perceptible  for  single  observations,  but  in  reducing  long  series  the  use 
of  a  system  of  weights  tends  to  diminish  the  influence  of  discordant 
comparisons,  and  to  smooth  out  final  results. 

While  it  is  not  possible  to  reproduce  here  any  star  maps  for 
the  observer  to  use,  enough  has  been  said  to  show  that  they  can 
very  easily  be  obtained  by  the  serious  observer,  either  from  the 
Harvard  Observatory,  or  from  the  secretary  of  the  Variable 
Star  Association.  The  latter  has  kindly  furnished  a  list  of  vari- 
ables that  are  easy  to  locate  because  of  their  proximity  to  lucid 
stars,  and  are  therefore  recommended  for  observation  with 
small  telescopes  which  are  unmounted.  In  the  list  is  given  the 
Harvard  number  as  well  as  the  name  of  the  star.  If  the  desig- 
nation is  underlined  it  signifies  that  the  star  is  of  south  declina- 
tion. The  second  list  contains  variables  recommended  for 
observation  to  those  who  have  small  telescopes  mounted,  with 
circles.  These  lists  are  not  in  any  sense  complete;  they  are 
merely  recommended  for  the  beginner. 

LIST  I 


072708 

S  Can.  Mm. 

092411 

R  Leonis. 

103769 

R  Urs.  Maj. 

141954 

S  Bootis. 

154428 

R  Cor.  Bor. 

162119 

U  Herculis. 

170215 

R  Ophiuchi. 

180531 

T  Herculis. 

184205 

R  Scuti. 

193449 

R  Cygni. 

230110 

R  Pegasi. 

LIST  H 

021403 

o  Ceti. 

023133 

R  Trianguli. 

043274 

X  Camelop. 

174922 

U  Geminorum. 

123160 

T  Urs.  Maj. 

123961 

S   Urs.  Maj. 

134440 

R  Can.  Ven. 

306         THE  STUDY  OF  VARIABLE  STARS 

142584  T  Camelop. 

154615  R  Serpentis. 

163266  R  Draconis. 

194632  x   Cygni. 

205923  R  Vulpeculae. 

210868  T  Cephei. 

213843  SS  Cygni. 

230759  V  Cassiopeiae. 

It  is  not  possible,  in  the  limits  of  this  volume,  to  include  an 
extended  bibliography  of  the  subject  of  variable  stars.  There 
are,  however,  a  few  publications  that  are  extremely  useful  and 
suggestive  to  the  non-professional  worker,  which  may  be  men- 
tioned here.  It  may  also  be  stated  that  Hagen,  in  the  first 
volume  of  Die  Verdnderlichen  Sterne,1  gives  a  very  good  bibli- 
ography which  contains  in  general  separate  books,  publications 
of  observatories,  and  proceedings  of  Academies,  but  not  ordi- 
nary articles  from  periodicals.  The  books  which  the  author  has 
found  to  be  specially  useful  are  Andre,  Traite  d'Astronomie 
Stellaire;2  Campbell,  Stellar  Motions;3  Clerke,  Problems  in 
Astro-Physics,*  and  The  System  of  the  Stars; 5  Scheiner,  Populare 
Astro-Physik,*  section  on  photometric  apparatus,  which  gives  a 
brief  account  of  a  great  many  of  the  different  kinds  of  photom- 
eters; Turner,  The  Great  Star  Map?  which  gives  an  account 
of  the  formation  of  the  Carte  du  del.  Gould's  Uranometria  Ar- 
gentina 8  is  excellent  for  its  miscellaneous  information  regarding 
magnitudes  and  star  maps;  Baly,  Spectroscopy,9  though  very 
technical,  has  some  useful  descriptive  and  historical  sections. 
Andre  has  the  best  material  on  the  history  of  stellar  magnitude. 
The  Annals  of  the  Harvard  Observatory  are  so  well  known  that 

Herdersche  Verlagshandlung,  Freiburg  in  Breisgau,  Germany  ;  1913. 

Gauthier-Villars,  Paris,  France;  1899. 

Yale  University  Press,  New  Haven,  Conn.;  1913. 

Adam  and  Charles  Black,  London;  1903. 

Longmans,  Green  and  Co.,  London  and  New  York;  1890. 

B.  G.  Teubner,  Leipzig  and  Berlin;  1908. 

E.  P.  Dutton,  New  York;  1912. 

Printed  by  Paul  Emile  Coni,  Buenos  Aires;  1879. 

Longmans,  Green  and  Co.,  London  and  New  York;  1905. 


HINTS  FOR  OBSERVERS  307 

it  is  hardly  necessary  to  make  any  special  reference  to  them. 
However,  they  contain  so  much  material  on  the  subject  of  vari- 
able stars  that  it  is  sometimes  difficult  to  find  exactly  what  one 
wishes.  For  the  purpose  of  obviating  this  difficulty  Pickering 
has  recently  issued  a  pamphlet  which  summarizes  the  contents 
of  the  Annals.  In  it  there  appears  first  a  classification  of  the 
volumes  according  to  the  subjects.  This  is  followed  by  a  list 
including  the  titles  of  the  different  parts  of  all  the  volumes. 
After  this  is  given  a  description  of  the  contents  of  each  part 
which  is  sufficiently  full  to  enable  the  investigator  to  find  the 
object  of  his  search.  The  writer  may  perhaps  mention  a  few  of 
the  volumes  which  have  proved  most  useful  in  her  work. 
Volume  37  was  the  first  to  give  the  lists  of  comparison  stars  for 
variables  and  the  method  of  determining  their  magnitudes. 
It  also  furnishes  the  method  of  determining  the  mean  light 
curve  of  long  period  variables.  The  second  part,  which  was 
issued  after  the  first  three  series  of  Hagen's  Atlas  had  appeared, 
contains  the  material  for  converting  the  Hagen  grades  into 
magnitudes  on  the  Harvard  photometric  scale,  for  many  stars 
in  Series  I-III. 

Volume  57  contains  also  observations  of  long  period  vari- 
ables, made  during  the  years  1902-1905;  the  second  part  in- 
cludes lists  of  comparison  stars  for  252  variables,  for  the  rest  of 
Hagen's  stars  in  Series  I-III,  not  included  in  Volume  37,  and 
also  stars  in  the  other  series.  Volume  55  contains  the  catalogue 
of  variable  stars,  also  a  table  stating  the  colors  of  variables,  so 
far  as  they  have  been  observed,  and  an  index  to  the  maps  which 
have  been  published.  Volume  24  contains  the  magnitudes  of 
20125  faint  stars,  in  zones  20'  wide,  and  having  centers  in 
decimations -20°,  -  15°,  -  10°,  etc.,  to  the  north  pole.  These 
are  particularly  useful  in  forming  light  scales  for  new  variables, 
where  the  observer  must  depend  upon  the  Durchmusterung 
maps  alone.  Volume  50  contains  the  magnitudes  of  9111  stars, 
mainly  of  mg.  6.5  and  brighter,  and  superseding  the  catalogue 
of  stars  in  the  preceding  volumes.  At  the  end  is  a  very  useful 
abbreviated  Table,  VII,  which  is  called  an  index  to  the  Bayer 


308          THE  STUDY  OF  VARIABLE  STARS 

and  Lacaille  letters.  The  stars  are  arranged  in  order  of  con- 
stellation, and  the  tables  contain:  first,  the  letter;  second,  the 
number  in  the  catalogue;  third,  the  magnitude;  and  fourth,  the 
spectral  type.  Volume  28  gives  the  classification  of  stellar 
spectra,  with  plates.  Volume  9  contains  Peirce's  work  with  the 
Zb'llner  photometer,  and  his  discussion  of  the  older  catalogues, 
including  the  manuscripts  of  Ptolemy's  Almagest. 


THE  END 


APPENDIX 


TABLE  I 


Table  la 
For  Century 

TaUe  Ib 
For  Yeat  in  Century 

Table  Ic 
For  the  Day 

-1900 

1  027  083 

00 

Oor-1 

50 

18  262 

-1800 

1  063  608 

01 

365 

51 

18  627 

-1700 

1  100  133 

02 

730 

52 

18  992 

Jan.  0 

0 

0 

-1600 

1  136  658 

03 

1  095 

53 

19  358 

10 

10 

10 

-1500 

1  173  183 

04 

1  460 

54 

19  723 

20 

20 

20 

30 

30 

30 

-1400 

1  209  708 

05 

1  826 

55 

20  088 

-1300 

1  246  233 

06 

2  191 

56 

20  453 

Feb.  0 

31 

31 

-1200 

1  282  758 

07 

2  556 

57 

20819 

10 

41 

41 

-1100 

1  319  283 

08 

2  921 

58 

21  184 

20 

51 

51 

-1000 

1  355  808 

09 

3287 

59 

21  549 

Mar.  0 

59 

60 

-  900 

1  392  333 

10 

3  652 

60 

21  914 

10 

69 

70 

-  800 

1  428  858 

11 

4  017 

61 

22  280 

20 

79 

80 

-  700 

1  465  383 

12 

4382 

62 

22  645 

30 

89 

90 

-  600 

1  501  908 

13 

4  748 

63 

23  010 

-  500 

1  538  433 

14 

5  113 

64 

23  375 

April  0 

90 

91 

10 

100 

101 

-  400 

1  574  958 

15 

5  478 

65 

23  741 

20 

110 

111 

-  300 

1  611  483 

16 

5  843 

66 

24  106 

-  200 

1  648  008 

17 

6  209 

67 

24471 

May  0 

120 

121 

-  100 

1  684  533 

18 

6  574 

68 

24  836 

10 

130 

131 

0 

1  721  058 

19 

6939 

69 

25  202 

20 

140 

141 

30 

150 

151 

+  100 

1  757  583 

20 

7  304 

70 

25567 

+  200 

1  794  108 

21 

7  670 

71 

25  932 

June  0 

151 

152 

+  300 

1  830  633 

22 

8  035 

72 

26  297 

10 

161 

162 

+  400 

1  867  158 

23 

8  400 

73 

26  663 

20 

171 

172 

+  500 

1  903  683 

24 

8  765 

74 

27  028 

July  0 

181 

182 

-f-  600 

1  940  208 

25 

9  131 

75 

27393 

10 

191 

192 

4-  700 

1  976  733 

26 

9  496 

76 

27  758 

20 

201 

202 

+  800 

2  013  258 

27 

9  861 

77 

28  124 

30 

211 

212 

4-  900 

2  049  783 

28 

10  226 

78 

28  489 

+1000 

2  086  308 

29 

10592 

79 

28  854 

Aug.  0 

212 

213 

10 

222 

223 

+1100 

2  122  833 

30 

10  957 

80 

29219 

20 

232 

233 

+1200 

2  159  358 

31 

11  322 

81 

29  585 

30 

242 

243 

+1300 

2  195  883 

32 

11  687 

82 

29  950 

+1400 

2  232  408 

33 

12  053 

83 

30  315 

Sept.  0 

243 

244 

+1500 

2  268  933 

34 

12  418 

84 

30  680 

10 

253 

254 

20 

263 

264 

+1600 

2  305  448 

35 

12  783 

85 

31  046 

+1700 

2  341  972 

36 

13  148 

86 

31  411 

Oct.  0 

273 

274 

+1800 

2  378  496 

37 

13  514 

87 

31  776 

10 

283 

284 

+1900 

2  415  020 

38 

13  879 

88 

32  141 

20 

293 

294 

+2000 

2  451  545 

39 

14244 

89 

32  507 

30 

303 

304 

+2100 

2  488  069 

40 

14  609 

90 

32  872 

NOV.  0 

304 

305 

+2200 

2  524  593 

41 

14  975 

91 

33  237 

10 

314 

315 

+2300 

2  561  117 

42 

15  340 

92 

33  602 

20 

324 

325 

+2400 

2  597  642 

43 

15  705 

93 

33  968 

44 

16  070 

94 

34  333 

Dec.  0 

334 

335 

10 

344 

345 

45 

16  436 

95 

34  698 

20 

354 

355 

46 

16  801 

96 

35  063 

30 

364 

365 

47 

17  166 

97 

35  429 

48 

17  531 

98 

35  794 

49 

17  897 

99 

36  159 

312 


APPENDIX 
TABLE  II 


da 

hm  t 

m  B 

8 

da 

h  m  s 

TO  a 

• 

0.01 

0  14  24 

0  8.64 

0.09 

0.51 

12  14  24 

1  20.64 

4.41 

0.02 

0  28  48 

0  17.28 

0.17 

0.52 

1228  48 

729.28 

4.49 

0.03 

0  43  12 

025.92 

0.26 

0.53 

12  43  12 

7  37.92 

4.58 

0.04 

0  57  36 

0  34.56 

0.35 

0.54 

12  57  36 

7  46.56 

4.67 

0.05 

1  12  0 

043.20 

0.43 

0.55 

13  12  0 

7  55.20 

4.75 

0.06 

1  26  24 

0  51.84 

0.52 

0.56 

13  26  24 

8  3.84 

4.84 

0.07 

1  40  48 

0.48 

0.60 

0.57 

13  40  48 

8  12.48 

4.92 

0.08 

1  55  12 

9.12 

0.69 

0.58 

13  55  12 

8  21.12 

5.01 

0.09 

2  9  36 

17.76 

0.78 

0.59 

14  9  36 

8  29.76 

5.10 

0.10 

2  24  0 

26.40 

0.86 

0.60 

14  24  0 

8  38.40 

5.18 

0.11 

2  38  24 

35.04 

0.95 

0.61 

14  38  24 

8  47.04 

5.27 

0.12 

2  52  48 

43.68 

1.04 

0.62 

14  52  48 

8  55.68 

5.36 

0.13 

3  7  12 

1  52.32 

1.12 

0.63 

15  7  12 

9  4.32 

5.44 

0.14 

3  21  36 

2  0.96 

1.21 

0.64 

15  21  36 

9  12.96 

5.53 

0.15 

336  0 

2  9.60 

1.30 

0.65 

15  36  0 

9  21.60 

5.62 

0.16 

3  50  24 

2  18.24 

1.38 

0.66 

15  50  24 

9  30.24 

5.70 

0.17 

4  4  48 

2  26.88 

1.47 

0.67 

16  4  48 

9  38.88 

5.79 

0.18 

4  19  12 

2  35.52 

1.56 

0.68 

16  19  12 

9  47.52 

5.88 

0.19 

433  36 

2  44.16 

1.64 

0.69 

16  33  36 

9  56.16 

5.96 

0.20 

4  48  0 

2  52.80 

1.73 

0.70 

1648  0 

10  4.80 

6.05 

0.21 

5  2  24 

3  1.44 

1.81 

0.71 

17  2  24 

10  13.44 

6.13 

0.22 

5  1648 

3  10.08 

1.90 

0.72 

17  16  48 

10  22.08 

6.22 

0.23 

5  31  12 

3  18.72 

1.99 

0.73 

17  31  12 

10  30.72 

6.31 

0.24 

5  45  36 

3  27.36 

2.07 

0.74 

17  45  36 

10  39.36 

6.39 

0.25 

600 

336.00 

2.16 

0.75 

18  0  0 

10  48.00 

6.48 

0.26 

6  14  24 

344.64 

2.25 

0.76 

18  1424 

10  56.64 

6.57 

0.27 

628  48 

3  53.28 

2.33 

0.77 

18  28  48 

11  5.28 

6.65 

0.28 

6  43  12 

4  1.92 

2.42 

0.78 

18  43  12 

11  13.92 

6.74 

0.29 

6  57  36 

4  10.56 

2.51 

0.79 

18  57  36 

11  22.56 

6.83 

0.30 

7  12  0 

4  19.20 

2.59 

0.80 

19  12  0 

11  31.20 

6.91 

0.31 

726  24 

427.84 

2.68 

0.81 

19  26  24 

11  39.84 

7.00 

0.32 

7  40  48 

4  36.48 

2.76 

0.82 

19  40  48 

11  48.48 

7.08 

0.33 

755  12 

445.12 

2.85 

0.83 

19  55  12 

11  57.12 

7.17 

0.34 

8  9  36 

4  53.76 

2.94 

0.84 

20  9  36 

12  5.76 

7.26 

0.35 

8  24  0 

5  2.40 

3.02 

0.85 

20  24  0 

12  14.40 

7.34 

0.36 

8  3824 

5  11.04 

3.11 

0.86 

20  38  24 

12  23.04 

7.43 

0.37 

8  52  48 

5  19.68 

3.20 

0.87 

20  52  48 

12  31.68 

7.52 

0.38 

9  7  12 

5  28.32 

3.28 

0.88 

21  7  12 

12  40.32 

7.60 

0.39 

9  21  36 

5  36.96 

3.37 

0.89 

21  21  36 

12  48.96 

7.69 

0.40 

9  36  0 

5  45.60 

3.46 

0.90 

21  36  0 

12  57.60 

7.78 

0.41 

9  50  24 

5  54.24 

3.54 

0.91 

21  50  24 

13  6.24 

7.86 

0.42 

10  4  48 

6  2.88 

3.63 

0.92 

22  4  48 

13  14.88 

7.95 

0.43 

10  19  12 

6  11.52 

3.72 

0.93 

22  19  12 

13  23.52 

8.04 

0.44 

10  33  36 

6  20.16 

3.80 

0.94 

22  33  36 

13  32.16 

8.12 

0.45 

10  48  0 

6  28.80 

3.89 

0.95 

22  48  0 

13  40.80 

8.21 

0.46 

11  2  24 

6  37.44 

3.97 

0.96 

23  2  24 

13  49.44 

8.29 

0.47 

11  16  48 

6  46.08 

4.06 

0.97 

23  16  48 

13  58.08 

8.38 

0.48 

11  31  12 

664.72 

4.15 

0.98 

23  31  12 

14  6  72 

8.47 

0.49 

11  45  36 

7  3.36 

4.23 

0.99 

23  45  36 

14  15.36 

8.55 

0.50 

12  0  0 

7  12.00 

4.32 

1.00 

24  0  0 

1424.00 

8.64 

APPENDIX  313 


EXPLANATION  OF  TABLES 

Table  I  is  for  the  purpose  of  converting  a  calendar  date  into  Julian 
Days  or  vice  versa.  It  is  divided  into  three  parts,  which  are  to  be  used 
as  follows:  la, for  the  beginning  of  the  century;  16,  for  the  year  in 
the  century,  and  Ic,  for  the  day  of  the  month.  Table  la  contains  the 
day  of  the  Julian  period  corresponding  to  the  first  day  in  each  cen- 
tury counted  according  to  the  Julian  calendar  through  1500,  but  ac- 
cording to  the  Gregorian  calendar  for  the  succeeding  centuries.  Table 
16  contains  the  number  of  days  in  each  year  of  the  century  counted 
from  its  beginning.  If  the  zero  year  of  the  century  is  a  leap  year,— 1 
should  be  used.  Table  Ic  gives  the  day  in  the  year  corresponding  to 
the  day  of  the  month.  It  contains  three  columns;  the  first  gives  the 
tenth  day  of  each  month;  the  second,  the  corresponding  day  of  the 
year  to  be  used  for  the  common  year;  and  the  third,  the  day  to  be 
used  for  a  leap  year.  The  use  of  the  table  can  best  be  explained  by 
an  example. 

Find  the  Julian  Day  corresponding  to  the  calendar  date  March  26, 
1915.  Entering  the  column  la  with  the  argument  1900,  the  corre- 
sponding number  will  be  2  415  020.  From  Table  16  find  the  number 
corresponding  to  15,  which  is  5478.  Since  1915  is  not  a  leap  year,  the 
day  of  the  year  for  March  26  should  be  taken  from  the  second  column 
of  Ic.  It  is  85.  The  sum  of  these  three  numbers  will  be  the  Julian  Day 
corresponding  to  the  given  calendar  date.  The  computation  may  be 
arranged  as  follows:  — 

March  26, 1915 

Table  la,  1900 2  415  020 

Table  16,   15 5  478 

Table  Ic,  March  26 85 


2  420  583 

The  reverse  process  is  frequently  necessary  in  the  prediction  of  the 
time  of  maximum  or  minimum  of  the  variable  star,  as  illustrated  in 
Chapter  XI.  The  computation  may  be  made  as  follows.  Given  Julian 
Day  2  405  693;  change  to  the  corresponding  calendar  date.  From 
Table  la  the  century  is  found  to  be  1800,  or  Julian  Day  2  378  496. 
Subtract  this  value  from  the  Julian  Day  given  and  the  result,  27  197, 
is  the  number  of  days  since  the  beginning  of  the  century.  From 
Table  16  this  is  found  to  be  74  years,  for  which  the  number  of  days  is 
27  028.  Subtracting  this  number  from  the  remainder  just  given,  the 


314  APPENDIX 

result,  169,  indicates  the  number  of  the  day  in  the  year.  Since  1874 
was  not  a  leap  year  this  number  should  be  taken  from  column  two  of 
Ic.  This  corresponds  to  June  18.  The  resulting  date  is,  then,  June  18, 
1874. 

Julian  Day 2  405  693 

Table  la,  1800 2  378  496 

Remainder 27  197 

Table  16,  74 27028 

Table  Ic,  June  18 169 

The  observer  who  is  working  continuously  with  the  observation  of 
variable  stars  will  often  find  it  convenient  to  arrange  a  small  table 
which  contains  the  zero  day  of  each  month  for  the  years  covered  by 
his  observations. 

Table  II  is  for  the  purpose  of  changing  from  hours,  minutes,  and 
seconds  to  the  fraction  of  a  day  or  vice  versa.  The  argument  of  this 
table  is  made  to  serve  for  all  three  columns  of  tabular  values.  As 
printed,  it  gives  the  first  two  decimal  places  of  the  fraction  of  a  day. 
The  tabular  values  for  this  are  given  in  hours,  minutes,  and  seconds 
in  the  second  column;  e.g.,  Od.a05  is  equivalent  to  lh  12m  0s.  The 
third  column  contains  the  minutes  and  seconds  for  .01  of  the  argu- 
ment; e.g.,  da.OO  05  is  Om  43?20.  The  numbers  in  the  fourth  column 
give  the  seconds  corresponding  to  .00  01  of  the  argument;  e.g., 
^.00  00  05  is  equivalent  to  0!43. 

For  example,  change  d?213  675  into  hours,  minutes,  and  seconds. 
With  the  argument  .21  take  the  value  from  the  second  column.  With 
the  argument  .00  36  take  the  value  from  the  third  column.  With  the 
argument  .00  00  75  take  the  value  from  the  fourth  column.  The  sum 
will  be  the  hours,  minutes,  and  seconds  corresponding  to  the  given 
fraction  of  a  day. 

Qda  21  5h  2m  248 

.00  36  5    11  .04 

.00  00  75  6  .48 


0.21  36  75  57    41 .52 

The  reverse  process  is  illustrated  by  the  following  example.  Change 
16h  23m  29!66  into  the  fraction  of  a  day.  From  column  two  we  find 
the  first  two  figures  of  the  result  to  be  Od.a68,  which  corresponds  to 
16h  19m  12s;  leaving  a  remainder  of  4m  IT.GG.  From  column  three  this 
is  found  to  correspond  to  d*00  29,  with  a  remainder  of  7?  10.  From 
column  four  this  is  found  to  correspond  to  da.OOOO  82,  which  is  equal 
to  7°08.  The  given  time  is  equivalent,  then,  to  .682  982  days. 


APPENDIX  315 

16h  23m  29S66 
Oda.68  16    19    12 


4     17.66 
.00  29  4     10.56 


7.10 
.00  00  82  7.08 


0.68  29  82 


DESCRIPTION  OF  STELLAR  SPECTRA 

The  following  three  plates  are  intended  to  illustrate  the  classifi- 
cation of  stellar  spectra,  which  was  explained  in  Chapter  I  of  this 
book,  being  abbreviated  from  the  original  in  H.C.O.,  Annals,  vol. 
28,  from  which  Plates  XIII  and  XIV  are  also  taken.  In  describing 
them,  an  attempt  will  be  made  to  indicate  the  groups  of  lines  on  which 
the  classification  is  based. 

The  first  plate,  No.  XII,  which  is  taken  from  Huggins's  Stellar 
Spectra,  represents  the  solar  spectrum,  which  is  of  Class  G,  and  four 
stars  of  Class  A.  The  photograph  covers  a  region  from  wave-length 
4050  to  wave-length  3625,  with  the  red  at  the  right.  As  the  K  line  at 
X  3933  is  at  the  limit  of  visibility,  the  greater  part  of  the  spectrum 
as  here  depicted  is  in  the  ultra-violet.  This  plate  is  particularly  val- 
uable because  it  shows  the  hydrogen  series  of  lines  (1),  of  Chapter  I, 
page  32.  The  lines  are  known  by  the  letters  of  the  Greek  alphabet,  the 
first  one  on  the  right  being  He,  and  can  be  followed  by  their  posi- 
tions as  far  as  HTT.  These  hydrogen  lines  are  always  recognized  in 
the  earlier  classes  of  spectra  by  their  rhythmic  order.  They  are  very 
apparent  in  types  A  and  F  of  Plate  XIII,  and  can  easily  be  followed 
into  B,  G,  and  K.  On  Plate  XIV  they  can  be  recognized  in  a  Cygni, 
f  Puppis,  and  o  Ceti.  In  the  last  star  they  are  bright  instead  of  dark. 

The  second  group  (2)  consists  of  another  series  of  hydrogen  lines, 
which  may  be  seen  in  the  spectrum  of  f  Puppis,  Plate  XIV,  where 
they  are  intermediate  between  the  hydrogen  lines,  a  little  to  the  left 
of  the  middle  of  each  space.  This  set  of  lines  is  characteristic  of  the 
early  stars. 

The  Orion  lines  (3),  which  comprise  the  lines  of  helium  and  other 
substances,  not  including  hydrogen,  are  seen  in  the  spectrum  of  e  Ori- 
onis.  It  should  be  stated  that  the  spectra  on  Plates  XIII  and  XIV 
include  the  lines  between  the  wave-lengths  3800  and  5000;  hence  they 


316  APPENDIX 

are  almost  entirely  in  the  visible  part  of  the  spectrum.  The  hydrogen 
lines  in  the  spectrum  of  a  Can.  Maj.  or  Sirius  are  thus  H/3,  Hy,  H5, 
and  He.  Nearly  all  the  lines  in  e  Orionis,  with  the  exception  of 
the  hydrogen  lines,  are  classed  as  characteristic  Orion  lines.  Of  the 
three  most  intense  lines  lying  between  H/3  and  Hy  the  first  and  last 
are  due  to  helium.  The  group  of  lines  near  H§  includes  an  oxygen 
triplet  to  the  left,  and  two  helium  lines  to  the  right.  The  strongest 
line,  which  is  nearly  midway  between  He  and  H§,  belongs  to  helium. 

The  calcium  lines  H  and  K  (4),  which  are  very  strong  in  the  solar 
spectrum,  appear  toward  the  right  of  Plate  XII,  where  they  have  a 
great  intensity,  and  also  on  Plate  XIII,  in  the  spectrum  of  a  Carinae, 
a  Aurigae,  and  a  Bootis.  H  coincides  very  closely  with  He,  He  hav- 
ing the  shorter  wave-length  of  the  two.  The  dispersion  is  not  suffi- 
cient in  any  of  the  accompanying  photographs  to  show  the  separa- 
tion between  the  two  lines.  The  K  line  appears  a  little  to  the  right 
of  the  center  of  the  space  between  He  and  Hf.  It  shows  as  a  rather 
faint  line  in  the  three  lower  spectra  of  Plate  XII,  where  it  is  not  as 
intense  as  the  hydrogen  lines. 

The  solar  lines  (5)  can  be  seen  readily  in  the  spectra  of  a  Aurigae 
and  a  Bootis,  on  Plate  XIII,  and  in  the  ultra-violet  portion  of  the 
spectrum  of  the  sun,  shown  in  Plate  XII.  It  would  be  impossible  to 
select  any  particularly  characteristic  lines  in  this  set,  but  they  may 
perhaps  be  described  as  being  of  rather  small  intensity  and  quite 
closely  crowded  together.  The  distinction  between  the  arrangement 
of  lines  in  Class  B  and  Class  G,  as  shown  on  Plate  XIII,  is  very  appar- 
ent. In  the  latter  star,  the  H  and  K  lines,  and  the  thickness  with 
which  the  solar  lines  are  packed  in  the  spectrum,  distinguish  it  at  once 
from  Class  B,  where  the  stronger  lines  are  isolated,  and  the  finer  lines 
are  scattered,  appearing  in  groups  of  not  more  than  two  or  three  lines 
together.  Group  G  (6)  appears  first  in  the  spectrum  of  a  Aurigae, 
just  a  little  to  the  left  of  Hy.  It  is  easily  recognized  because  it  con- 
sists of  two  rather  strong  lines  in  the  midst  of  a  large  number  of  fine 
lines.  This  group  is  one  of  the  distinguishing  features  of  Class  G. 

The  bright  bands  (7)  are  seen  in  the  spectrum  of  y  Velorum,  Plate 
XIV. 

It  is  important  to  note  how  the  different  sets  of  lines  just  described 
change  in  intensity  in  passing  from  one  spectral  type  to  another,  and 
this  may  be  traced,  to  a  certain  extent,  on  Plate  XIII,  though  there 
are  not  enough  examples  of  spectra  given  there  to  show  it  in  detail. 
The  broad  hazy  bright  bands  appear  first,  and  are  accompanied  by 
bright  hydrogen  lines  of  both  series.  The  star  y  Velorum  has  the 
bright  bands,  but  the  hydrogen  lines  are  dark.  For  this  reason  it  is 
marked  Oa  Pec.,  for  if  it  were  a  typical  star  of  this  class,  the  lines 


APPENDIX  317 

would  be  bright  instead  of  dark.  The  hydrogen  lines  become  nar- 
rower and  finally  give  place  to  dark  hydrogen  lines  of  both  series,  as 
illustrated  in  the  spectrum  of  f  Puppis.  The  series  (1)  increases  in  in- 
tensity through  Class  B,  reaching  its  maximum  in  Class  A,  as  illus- 
trated by  the  spectrum  of  Sirius.  They  then  diminish  through  Class  F 
and  G  into  K,  in  which  they  are  no  more  conspicuous  than  many  other 
solar  lines.  The  second  series  of  hydrogen  lines  reaches  its  maximum 
intensity  in  Od,  and  then  quickly  disappears.  They  can  also  be  seen 
faintly  in  the  spectrum  of  7  Velorum. 

The  Orion  lines  appear  as  the  bright  bands  (7)  disappear,  increase 
in  intensity,  reaching  a  maximum  in  B  2  and  B  3,  then  diminish,  and 
in  A  are  barely  visible,  c  Orionis  belongs  to  Class  B,  in  which  the 
Orion  lines  have  not  yet  reached  their  greatest  intensity.  The  cal- 
cium line,  K,  appears  in  Class  A,  as  pointed  out  in  the  spectrum  of 
Sirius,  and  increases  in  intensity  until,  with  H,  it  dominates  the  spec- 
trum. The  other  solar,  or  metallic,  lines,  appear  faintly  in  type  A, 
become  more  and  more  strengthened,  particularly  the  G  band,  and 
finally  the  entire  spectrum  becomes  banded.  Attention  should  be 
called  to  the  spectrum  of  o  Ceti,  the  well-known  long  period  variable, 
because  its  spectrum  is  of  the  banded  type,  having  bright  hydrogen 
lines  at  maximum.  It  is  interesting  to  note  that  these  bright  lines 
have  not  all  the  same  intensity.  Those  seen  in  the  illustration  are 
H/3,  H>y,  Hs,  Hf,  and  Hi;.  H€  is  absent.  The  explanation  given  is  that 
it  is  hidden  by  an  over-lying  calcium  line,  H.  If  the  reader  is  inter- 
ested in  studying  the  spectra  of  the  stars  further,  ample  explanation 
for  identifying  the  lines  will  be  found  in  volume  28,  referred  to  above. 


B 

>* 

H 

a  § 

^     **< 

5     g 
^     £ 


A  I 


FI-II 


G  II 


K  II 


Malll 


ORIONIS 


a  CAN.  M. 


a  CAKINAI 


a  AURIGAI 


a  BOOTIS 


a  ORIONIS 


Plate  XIII 

TYPICAL  SPECTRA 


A2F  Pec. 


Oa  Pec. 


Od 


BPec. 


Md. 


a  CYGNI 


7  VELORU 


PUPPIS 


7CASSIOPE 


A*  CENTAU; 


o  CETI 


Plate  XIV 

PECULIAR  SPECTRA 


INDEX 


INDEX 


Algol,  type,  6,  13,  17,  241 ;  description  of 
type,  14;  light  curve,  15;  spectrum  of 
type,  36;  on  Hagen  Charts  Series  IV,  58; 
Ephemeris  in  Hartwig,  72 ;  discovery  of 
variable  by  Goodricke,  75,  269;  method 
of  observing  with  selenium  photometer, 
156,  157;  discovery  of  secondary  mini- 
mum, 157;  use  of  mean  light  curve  to 
correct  for  asymmetry,  199,  200;  expla- 
nation of  variability  of  type,  228-236; 
Goodricke's  hypothesis,  228, 229;  dimen- 
sions of  system,  229,  230;  Stebbins,  230; 
displacement  of  lines  in  spectrum,  233; 
normal  position  of  lines  at  minimum, 
235;  connection  between  orbital  motion 
and  displacement  of  lines,  236;  par- 
tial eclipse  shown  by  short  minimum, 
239. 

Allen,  photo-electricity,  158-159. 

Almagest,  81-82. 

Al  Sufi,  magnitudes,  82-83. 

American  Association  of  Variable  Star 
Observers,  296-297, 305. 

Anderson,  discovery  of  new  stars,  6,  75. 

Andre,  306 ;  method  of  naming  variables, 
69-70. 

Angstrom  unit,  21. 

Ant-Algol  stars,  13,  72. 

S  Arae,  13, 14. 

Argelander,  Uranometria  Nova,  38,  39, 
88,  89, 92, 126 ;  Banner  Durchmusterung, 
40,  41;  second  edition,  41;  director  of 
Observatory  at  Abo,  42;  detailed  ac- 
count of  Durchmusterung,  42,  43,  44; 
limiting  magnitude,  45;  arrangement 
of  stars,  45;  description  of  catalogue, 
45;  description  of  charts,  46, 47,48;  con- 
tinuation of  Durchmusterung  to  —23° 
by  Schonfeld,  49;  use  of  precession,  51; 
method  of  determining  positions  of 
stars,  57;  method  of  naming  variables, 
68;  variables  discovered  in  making  cat- 
alogue, 76;  Bayer's  method  of  assigning 
letters,  83;  William  Herschel's  method 
of  determining  magnitude,  86;  Arge- 
lander step  method,  86, 170,  256,  269, 289, 
290, 292, 294 ;  DM.  magnitudes,  87-88,  89- 
91 ;  cases  of  variability  in  Durchmus- 
terung, 91 ;  scale  of  Uranometria  Nova, 
93;  description  of  step  method,  103-106; 


modifications  of  step  method,  107-108; 
DM.  magnitudes  used  as  standards, 
121-122;  color  scale,  130;  effect  on  color 
of  increased  aperture,  131 ;  step  method 
adapted  by  Chandler  to  color  scale,  132 ; 
step  method  adapted  to  photographic 
magnitudes,  153;  notation  of  magni- 
tudes, 168;  sketch  of  life,  257-260;  suc- 
ceeded by  Schonfeld,  260;  friendship 
with  Heis,  262;  friendship  with  Krue- 
ger,  264. 

SS  Aurigae,  10. 

Averted  vision,  use  in  observing  faint 
stars,  104. 

Bailey,  discovery  of  variables  in  cluster 
<o  Centauri,  80. 

Baly,  Spectroscopy,  306. 

Barnard,  observations  of  temporary  stars, 
8-9;  color  of  Nova  Lacertae,  134-135; 
Bruce  photographic  telescope,  144. 

Baxendell,  observations  of  U  Gemino- 
rum,  250-251;  sketch  of  life,  266-267. 

Bayer,  position  of  Mira  Ceti,  74;  Urano- 
metria, 83-85,  89. 

Becker,  suspected  variables,  77. 

Beljawsky,  statistical  study  of  color  and 
period,  275-278  ;  period  and  range,  278. 

Bell,  color  of  double  stars,  135-137. 

Bemporad,  use  of  extinction  photometer, 
129. 

Bessel,  effort  to  construct  star  charts, 
41-42. 

Bibliography,  brief,  306-307. 

Slink  Stern,  substitution  for  "  Cepheid," 
12, 13. 

Blue  stars,  134. 

Bond,  study  of  photographic  image,  137- 
138. 

British  Association  of  Variable  Star  Ob- 
servers, 266,  298. 

Brooks,  editor  of  Pogson's  observations, 
265 ;  director  of  British  Association  of 
Variable  Star  Observers,  266,  298. 

Campbell,  components  of  spectroscopic 
binary,  226;  elements  of  orbit,  227, 
306. 

Campbell,  Leon ;  discussion  of  SS  Cygni, 
251 ;  observer  at  Harvard,  295. 


322 


INDEX 


Cannon,  classification  of  stellar  spectra, 
31-34,37;  Argelander's  Appeal  toFriends 
of  Astronomy,  258. 

ij  Carinae,  peculiarities  of  spectrum,  252- 
253. 

Carte  du  del,  charts,  51. 

Catalogues  of  variables,  Harvard  provi- 
sional, 5,  73;  Harvard,  51,  70,  77;  Hart- 
wig's,  70-72 ;  Chandler's,  73. 

S  Cephei,  light  curve  typical  of  Class  IV, 
11;  spectrum  of  type,  36;  formation  of 
light  scale  by  Schonf  eld,  170 ;  explana- 
tion of  type,  227 ;  spectroscopic  binary, 
244 ;  Shapley's  theory,  246, 247 ;  discovery 
of  variability  by  Goodricke,  269. 

U  Cephei,  light  curve,  16 ;  discovery  of 
variability,  76,  77;  correction  for  asym- 
metry of  light  curve,  200,  201 ;  long  min- 
imum, 239. 

Cepheid  variables,  typical  star,  11 ;  Blink 
Stern,  12, 13 ;  spectrum,  36 ;  theories  in 
explanation  of  type,  227  ;  spectroscopic 
binaries,  244;  Shapley's  theory,  246; 
cluster  type  identical  with  Cepheid, 
246;  variables  with  periods  less  than 
day,  247. 

Ceraski,  Prof,  and  Mme.,  discovery  of 
variability  of  U  Cephei,  76-77. 

Chandler,  material  in  Hagen  Charts,  53, 
54,  57,  58,  62,  63  ;  method  of  numbering 
variables,  69;  catalogues,  73;  color 
scale,  130;  method  of  forming  scale, 
131-132;  correction  for  asymmetry  of 
light  curve,  201 ;  elements  in  Hartwig, 
212 ;  M-m,  212;  obituary  of  Gould,  270, 
271 ;  sketch  of  life,  272 ;  colors  of  stars 
based  on  Chandler's  scale,  275. 

Charlier,  formation  of  photographic  star 
image,  140;  method  of  determining  stel- 
lar magnitudes  from  measurement  of 
photographic  images,  145-147. 

Classification  of  variable  stars,  Picker- 
ing's, 5-17. 

Clerke,  temporary  stars,  6,  253,  306;  Ce- 
pheid system,  245;  rj  Carinae,  252;  Good- 
ricke, 269. 

Cluster  type  of  variable,  14;  distinguished 
from  Cepheid  variable,  246-247. 

Color,  star;  relation  to  spectrum,  134-135, 
148-151;  Nova  Lacertae,  Barnard,  134- 
135;  of  double  stars,  135-137;  effect  on 
photographic  image,  137,  145,  148-151; 
of  variables,  287-288. 

Color  curve,  141-142. 

Color  intensity,  149;  and  spectral  type, 
Parkhurst  and  Jordan,  149-150;  King, 
151. 

Color  scale,  Potsdam  Photometric  Durch- 
musterunff,  121, 130;  Schmidt's,  130-131 ; 


effect  of  instrument  on,  131;  Chandler's' 
131-132;  Osthoff's,  132-133. 

Color  screen,  142. 

Comparison  spectrum,  218,  221. 

Cordoba  Durchmusterung,  49-50,  57. 

Cornu,  see  Hartmann-Cornu. 

R  Coronae  Borealis,  252,  270. 

Correcting  lens,  141-142,  221-222. 

Curtiss,  study  of  0  Lyrae,  243-244;  theory 
of  Cepheid  type,  245-246. 

SS  Cygni,  class,  6;  Parkhurst's  light 
curve,  10;  discussion  of  light  curve  by 
Leon  Campbell,  251 ;  diagram  from  Pop- 
ular Astronomy,  251-252. 

Discovery  of  variable  stars,  6 ;  announce- 
ment, 68;  methods,  73-80;  visual,  73-78; 
photographic,  78-79, 80;  from  spectrum, 
79-80;  miscellaneous,  265,  286;  earliest 
made,  286-287. 

Doppler-Fizeau  principle,  28;  explana- 
tion, 217-218. 

Double-slide  plate-carrier,  for  photog- 
raphy, 139. 

Double  stars,  colors  of,  135-137. 

Dove's  phenomenon,  108, 109. 

Duncan,  explanation  of  Cepheid  vari- 
ables, 246. 

Durchmusterung,  charts  for  telescopic 
work,  38,  39;  division,  40;  detailed  ac- 
count, 41, 42;  method  of  Argelander,  43, 
44;  present  use  of  original  records,  44; 
limiting  magnitude,  45 ;  arrangement  of 
stars  in  catalogue,  45;  description  of 
chart,  46-47;  second  edition,  48;  contin- 
uation of  Durchmusterung  to  —23°  by 
Schonf  eld,  49;  connection  with  Hagen 
charts,  54,  57,  58 ;  scale  of  magnitude, 
56;  enlargements  of  charts,  65-66;  con- 
nection with  Harvard  catalogue,  70; 
connection  with  Ephemeris,  71 ;  magni- 
tudes, 89-91 ;  connection  with  Potsdam 
Durchmusterung,  121,  122;  with  Har- 
vard Photometry,  126. 

Dlirer,  Albrecht,  drawings  for  Bayer's 
Uranometria,  84. 

Eclipsing  binary,  general  relations,  230; 
light  curve.  230;  relative  orbit,  230-231; 
real  orbits,  232 ;  spectroscopic  evidence, 
233;  velocity  curve,  233,  234;  conclu- 
sions, 235-236 ;  possible  combinations  in 
system,  236-239. 

Elements  of  variation,  4-5. 

Elster  and  Geitel,  studies  in  photo-elec- 
tricity, 158-159,  163. 

Enebo,  discovery  of  Nova  Geminorum  no. 
2,  254;  elements  of  SW  Geminorum, 
275. 


INDEX 


323 


Espin,  discovery  of  variable,  76. 
Extra-focal  images,  151-152. 

Fabricius,  first  observation  of  Mira  Ceti, 
74. 

Fechner,  psycho-physical  law,  and  con- 
nection with  stellar  magnitude,  98,  99. 

Fizeau,  see  Doppler-Fizeau. 

Fleming,  discovery  of  long  period  vari- 
able, 79;  discovery  of  new  star,  79-80; 
discovery  of  variable  in  cluster  Messier 
5,  80;  list  of  stars  of  fifth  type,  134;  sub- 
divisions of  spectra,  Class  M,  278-279. 

Fowler,  spectrum  of  hydrogen  and  heli- 
um, 37. 

Franks,  spectral  type  and  color,  282. 

Fraunhofer,  stellar  spectra,  30. 

Frost,  loss  of  light  in  spectograph,  223. 

Galactic  distribution  of  variables,  282- 
284. 

£  Geminorum,  14. 

IJ  Geminorum,  Class  ITb,  6,  250;  descrip- 
tion of  variation,  10,  250;  observations 
edited  by  Turner,  266,  267,  268. 

Gill,  78,  252. 

Goetz,  drawings  for  Hagen  charts,  62. 

Goodricke,  discovery  of  variability  of  /3 
Lyrae,  75;  explanation  of  variability  of 
Algol,  228-229;  sketch  of  life,  269. 

Gould,  Uranometria  Argentina,  62,  63, 
91-93, 126,  306;  sketch  of  life,  270-272. 

Grover,  observer  at  Sir  Cuthbert  Peek's 
observatory,  268. 

Guiding  telescope,  138-139. 

Guthnick  and  Prager,  photo-electric 
photometer,  154,  159-167. 

Hagen,  star  charts  for  variables,  52-53; 
designation  in  light  scale,  172;  light 
scale  for  Heis's  comparison  stars,  179; 
editor  of  Heis's  observations,  186,262; 
reproduction  of  Pogson's  charts,  266; 
bibliography,  306;  statement  regarding 
colorimetry,  134. 

Hagen  charts,  description  of  Series  I-III, 
52-58;  description  of  Series  IV,  58-61; 
magnitudes  of  Series  IV,  59-61 ;  descrip- 
tion of  Series  V,  61-64;  charts  by  Goetz, 
62;  description  of  Series  VI,  64,  65;  gen- 
eral index,  65 ;  index  to  Parkhurst  pho- 
tographs, 67;  use,  170,  293,  304,  307. 

Halma,  M.  1'Abbe,  edition  of  Almagest, 
81. 

Harding,  discovery  of  several  long  period 
variables,  265. 

Hartmann,  sensitiveness  of  photographic 
plate,  138;  -Cornu  formula,  224. 

Hartwig,  adoption  of  name  Blink  Stern 


for  Cepheid  variables,  12,  71 ;  classifica- 
tion of  S  Cephei  and  S  Arae,  13;  stars  of 
£  Geminorum  type,  14;  minima  of  Al- 
gol type,  62;  yearly  Ephemerides,  62, 68, 
70-72,  204,  208,  212,  215,  273,  283,  286;  pe- 
riod of  8  Cephei,  192;  elements  of  Al- 
gol, 212 ;  formula  for  reduction  to  sun, 
213;  stars  in/3  Lyrae  class,  241;  period 
of  S  Ursae  Majoris,  248;  elements  of 
SW  Geminorum,  275. 

Harvard  College  Observatory,  contents 
of  Annals  pertaining  to  variable  stars, 
306-308. 

Harvard  photographs,  used  to  trace  his- 
tory of  new  star,  6,  8;  maps  for  vari- 
able star  observers,  65,  66 ;  connection 
between  spectral  type,  color,  and  mag- 
nitude, 136 ;  variation  of  size  of  image, 
137, 138;  relation  between  spectral  type 
and  color  intensity,  King,  151-152;  mag- 
nitude determined  by  photographic  im- 
ages, 152, 153. 

Harvard  Photometry,  magnitude  in 
Schurig's  Atlas,  40 ;  magnitudes  in  Ha< 
gen,  Series  IV,  59;  magnitudes  in  Ha- 
gen, Series  V,  63;  magnitudes  in  Hagen, 
Series  VI,  64;  magnitudes  obtained  for 
757  stars,  82;  publication  of  first  vol- 
ume, 126;  Revised  Harvard  Photome- 
try, 173. 

Heis,  Atlas  Coelestis,  38,  39,  62,  126; 
adopted  Uranometria  Nova  as  model, 
89;  order  in  light  scale,  172;  use  of  ob- 
servations, 179,  186;  collected  observa- 
tions, 256 ;  friendship  with  Argelander, 
258 ;  sketch  of  life,  262-263. 

u  Herculis,  10, 11,  36. 

Herschel,  John,  numerical  relation  be- 
tween magnitude  and  relative  bright- 
ness, 94-95,  97. 

Herschel,  William,  method  of  comparing 
stellar  brightness,  85-87;  fractional 
magnitudes,  88 ;  importance  of  magni- 
tude, 94,  167;  comparison  of  method 
with  Argelander's,  103;  fluctuating 
stars,  105,  219. 

Hertz,  effect  of  light  on  electric  dis- 
charge, 158. 

Hind,  discovery  of  U  Geminorum,  250; 
interest  in  variable  stars,  267. 

Hipparchus,  globe  of,  81-82;  star  cata- 
logue, 94, 167. 

Hisgen,  drawings  for  Hagen  charts,  58 ; 
grades  for  Hagen  charts,  59. 

Holwarda,  discovery  of  variability  of 
Mira  Ceti,  74-75. 

Huggins,  early  work  in  spectroscopy, 
218-219. 

RV  Hydrae,  magnitude  curve  for,  61. 


324 


INDEX 


Irregular  variables,  color,  range,  252-263, 
284-285;  spectral  type,  285. 

Johnson,  determination  of  magnitude  ra- 
tio, p,  96,  98. 

Jordan,  see  Parkhurst  and  Jordan, 
Julian  Day,  5,  70;   explanation  of,  204- 
206;  table  for  converting  into  calendar 
date,  311,  313. 

Kempf,  see  MUller  and  Kempf. 

King,  color  intensity  and  spectral  type, 

150. 

Kirchhoff 's  law,  23. 
Klein,  star  atlas,  38. 
Knott,  observations  of  U  Geminorum, 

250-251 ;  collected  observations,  257, 265; 

sketch  of  life,  266,  267-268. 
Krueger,   assistant   to  Argelander,  48; 

catalogue  of  colored  stars,  63,  64;  DM. 

magnitudes,  90 ;  collected  observations, 

256,  263 ;  sketch  of  life,  264. 
Kustner,  present  use  of  Durchmuster- 

ung  records  at  Bonn,  44, 48,  76,  77. 

Lalande,-ffis<oire  Celeste,  41 ;  magnitudes 
in,  88. 

Least  Squares,  statement  of  principle, 
191-192. 

Light  curve,  single,  3-4 ;  method  of  plot- 
ting, 174;  determination  of  period,  174- 
176;  of  maximum,  176. 

Light  curve,  mean,  5  ;  plotting  of  obser- 
vations, 186-188 ;  approximate  elements, 
188;  correction  of  approximate  ele- 
ments, 188-190,  192-193;  determination 
of  phase,  193-194 ;  arrangements  of  ob- 
servations according  to  phase,  194 ; 
grouping  of  observations,  194-196; 
mean  light  curve,  196-197;  recapitula- 
tion, 198-199;  use  of  curve  for  Algol 
type,  Yendell's  method,  199-201;  Chand- 
ler's, 201-202 ;  Harvard  modification  of 
method  for  long  period  variables,  202- 
203. 

Light  scale,  formation  of,  170-173;  change 
from  light  step  into  magnitude  by  for- 
mula, 173-174;  by  curve,  177-179;  Park- 
hurst's  method  with  weights,  180-185. 

Light  wave,  19 ;  length,  19-20 ;  connection 
with  color,  20-21. 

List  of  variables  for  beginners,  305-306. 

Lockyer,  enhanced  lines,  26. 

Long  period  variables,  irregularities  in 
range  and  period,  248-249;  type  of  spec- 
trum, 248,  249-250 ;  color,  250 ;  subdivi- 
sions in  Class  II,  250-252. 

Lynn,  sketch  of  Goodricke's  life,  269. 

ft  Lyrae,  typical  star  of  Class  IVb,  6, 13; 


light  curve,  13;  Algol  type,  17,  241; 
spectrum  of  type,  36;  Ephemeris,  Hart- 
wig,  Table  IV,  72;  discovery  by  Good- 
ricke,  75,  269;  account  of  spectrum, 
242-244;  observations  by  Heis,  262;  flat- 
tening of  disc,  286. 

Magnitude  curve,  Hagen  grades  and 
Harvard  magnitudes,  61. 

Measurement  of  displacement  of  spectral 
lines,  223-225. 

Meridian  photometer,  description,  122- 
124;  method  of  reduction,  124-125. 

Meyer  and  Rosenberg,  photo-electric  pho- 
tometer, 154. 

Mira  Ceti,  discovery,  74-75;  light  curve 
and  maximum,  176-179;  irregularities 
in  period  and  magnitude,  248-249;  spec- 
trum, 249-250. 

Moore,  spectrum  of  y  Carinae,  252-253. 

Mt.  Wilson  Solar  Observatory,  Labora- 
tory of,  28. 

Muller,  Committee  of  Astronomische 
Gesellschaft,  65. 

Muller  and  Kempf,  PotsdamPhotometric 
Durchmusterung,  120-122;  magnitudes 
of  Pleiades,  299. 

Nova  Aurigae,  discovery  by  Anderson,  6; 
account  of  discovery,  6,  8. 

Nova  Geminorum  no.  2,  change  in  spec- 
trum, 35,  254. 

Nova  Lacertae,  Barnard,  color  of,  134-135. 

Nova  Persei,  discovery  by  Anderson,  6, 
75;  light  curve,  6, 253;  history,  8;  change 
of  spectrum,  35,  254;  theories,  253,  254. 

Observation  of  variables,  precautions, 
104-105,  108-110,  290-291,  303-305;  distri- 
bution of,  292-293;  methods,  294. 

Observations,  collected,  Argelander, 
257-260;  Schbnfeld,  260-262;  Heis,  262- 
264;  Schmidt,  264-265;  Pogson,  265-266; 
Baxendell,  266-267;  Knott,  267-268; 
Peek,  268-269. 

Observatories  co-operating  in  radial  ve- 
locity work,  225. 

Olcott,  secretary  American  Association 
of  Variable  Star  Observers,  297,  298, 
302,  305. 

a  Orionis,  10. 

Osthoff,  color  scale,  63,  132-133,  275,282; 
color  of  faint  stars,  133,  277;  color  of 
doubles,  135. 

Palisa  and  Wolf,  photographic  charts,  50. 
Paris  Ecliptic  Charts,  38,  50-51,  53. 
Parkhurst,  light  curve  of  SS  Cygni,  10; 
photographs  of  Hagen  fields,  66-€7;  maps 


INDEX 


325 


in  Researches  in  Stellar  Photometry, 
67;  researches  with  photographic 
wedge  photometer,  128;  formula  for 
photographic  magnitude,  147;  order  in 
light  scale,  172;  formation  of  light  scale 
with  weights,  180-185;  minimum  of  V 
Delphini,  275;  magnitudes  of  very  faint 
stars,  299;  precautions,  303-304. 

Parkhurst  and  Jordan,visual  magnitudes 
obtained  photographically,  148-149;  rela- 
tion between  color  intensity  and  spec- 
tral type,  149-150;  method  of  extra  focal 
images,  152. 

Peek,  observatory  at  Lyme  Regis,  265, 


Peirce,  study  of  Ptolemy's  magnitudes, 
82;  DM.  magnitudes,  88,  89-91. 

Perrine,  Cordoba  jDurchmusterung,  49- 
50;  investigation  of  photographic  im- 
ages, 140. 

Peters,  star  charts,  51,  53. 

Phase,  in  light  curve,  definition,  193. 

Photo-electricity,  principles  of,  158-159; 
applied  to  photometry,  159-167. 

Photographic  image,  affected  by  star 
color,  137,  144-145,  148-151;  appearance 
and  formation  of  image,  138-145;  theory 
of  Charlier,  140;  experiments  of  Perrine, 
140;  appearance  due  to  kind  of  tele- 
scope, 140-141 ;  effect  of  chromatic  aber- 
ration, 141-142;  use  of  color  screen,  142; 
use  of , stained  plates,  142, 148-151 ;  effect 
of  focal  length  of  telescope,  142-144; 
effect  of  spectral  type,  145. 

Photographic  magnitude,  method  of  de- 
termining, Charlier,  145-147;  Park- 
hurst,  147;  method  of  extra  focal  im- 
ages, 151-152;  Harvard  method,  152-153. 

Photographic  telescope,  Bruce,  144. 

Photographic  wedge,  method  of  prepar- 
ing, 151. 

Photography,  importance  in  spectro- 
scopic  work,  219. 

Photometer,  Zollner,  description,  116-120; 
—  use  in  preparing  Potsdam  Photo- 
metric Durchmusterung,  120,  122;  me- 
ridian, 122-126;  photographic  wedge, 
126-129;  extinction,  129;  selenium,  154- 
158;  photo-electric,  154, 158-167. 

Photometry,  development  of,  167-169. 

Pickering,  classification  of  variables,  5; 
subdivisions  of  Class  IV,  11;  Picker- 
ing's Type  V,  30;  secondary  hydrogen 
series,  31, 37;  Harvard  photometric  mag- 
nitude adopted  by  Hagen,  60,  61,  64; 
elements  used  by  Hagen,  65;  photo- 
graphic maps  for  amateur  observers, 
66;  variation  of  TY  Aquilae  confirmed 
by,  77;  discovery  of  SV  Herculis,  77-78; 


study  of  Orion  region  with  photo- 
graphic plates,  78;  discovery  of  long 
period  variables  with  bright  H  lines, 
79;  discovery  of  variable  in  Omega 
Centauri,  80;  comparison  of  Ptolemy's 
and  Harvard  magnitudes,  82;  publica- 
tion of  Herschel's  manuscript,  87;  dis- 
cussion of  Herschel's  magnitudes,  87; 
method  of  comparing  variables,  106-107; 
method  of  direct  estimation  of  magni- 
tudes, 107;  meridian  photometer,  122- 
126;  variation  of  Polaris,  125;  photo- 
graphic wedge  photometer,  126-129; 
re'sume'  of  methods  in  photographic 
photometry,  153;  explanation  of  vari- 
ability of  Algol,  229;  spectrum  of  ft 
Lyrae,  243 ;  subdivisions,  in  Class  II,  250 ; 
Class  He,  252;  publication  of  Arge- 
lander's  observations,  260;  publication 
of  Schonf eld's  observations,  262;  pub- 
lication of  Schmidt's  observations,  264; 
circulars  concerning  variable  star  ob- 
servations, 289-296,  303;  Section  of  Va- 
riable Star  Observers,  297-298. 

Pigott,  sketch  of  work,  269,270;  friend- 
ship with  Goodricke,  269. 

Pogson,  rule  for  expressing  relation  be- 
tween magnitude  and  relative  bright- 
ness, 94, 95, 96, 97 ;  connection  with  Fech- 
ner's  law,  99;  examples  of  rule,  100-102; 
application  in  observation,  106;  appli- 
cation of  rule  to  Algol,  229 ;  collected 
observations,  257,  266 ;  sketch  of  life, 


Polaris,  variation  in  brightness,  125. 

Polarized  light,  application  to  photom- 
etry, 114-116. 

Potsdam  Photometric  Durchmusterung, 
description,  120-122;  color  scale,  121, 130; 
40,  57,  61,  63. 

Prager,  see  Guthnick  and  Prager. 

Precession,  for  Durchmusterung  cata- 
logue, 51. 

Prediction  of  maximum  and  minimum  of 
variable  stars,  206-207;  method  of  find- 
ing E,  207-208;  examples  of  method  for 
long  period  variables,  208-211 ;  for  Algol, 
212-213. 

Ptolemy,  description  of  catalogue,  81-83; 
fractional  magnitudes,  82,88,  167,  168; 
magnitudes  adopted  by  later  astron- 
omers, 82-83;  value  of  division  into 
magnitudes,  82;  division  into  magni- 
tudes, 94,  97,  99. 

Purkinje  phenomenon,  108-109, 131, 137. 

Radial  velocity,  definition^?;  measure- 
ment of  ,"223-224;  observatories  engaged 
in  measuring,  225. 


INDEX 


Red  stars,  precautions  in  observing,  104- 
105,  108. 

Reduction  to  sun,  213-215. 

Refraction,  description  of  principle,  110- 
111;  double  refraction,  111-114;  applica- 
tion to  photometry,  114-116. 

Roberts,  magnitudes  for  southern  vari- 
ables, 63. 

Rosenberg,  see  Meyer  and  Rosenberg. 

Rowland's  tables  of  solar  wave-lengths, 
224-225. 

Russell,  effect  of  duration  of  minimum 
on  Algol  system,  239. 

Russell  and  Shapley,  hypothesis  of  dark- 
ened limb  in  Algol  variables,  240. 

Rutherford,  early  work  in  spectroscopy, 
219. 

Scheiner,  306;  early  work  in  spectroscopy, 
219. 

Schmidt,  color  scale,  63,  64,  130-131,  132, 
133;  connection  between  color  and 
period  of  variables,  132 ;  color  of  doubles, 
135;  collected  observations,  257;  sketch 
of  life,  264-265. 

Schb'nfeld,  assistant  to  Argelander,  48; 
southern  Durchmusterung,  49;  cata- 
logue of  variables,  57, 73;  magnitudes  in 
DM.,  88,  89-91;  collection  of  observa- 
tions, 170,  256;  order  in  light  step,  172; 
estimate  of  Argelander,  259;  sketch  of 
life,  260-262. 

Schultz,  photo-electric  photometer,  154, 
158. 

Schurig,  star  atlas,  38,  39-40. 

Secchi,  classification  of  stellar  spectra, 
30;  early  spectroscopic  work,  219. 

Seeliger,  galactic  distribution  of  stars, 
283-284. 

Selenium  cell,  153, 154;  physical  principle, 
154-155;  description,  155-158;  value  for 
short  period  variables,  157, 168,  229. 

Shapley,  relative  luminosity  of  compo- 
nents of  eclipsing  binary,  240;  spectral 
types  of  eclipsing  binaries,  241;  "A 
Study  of  the  Orbits  of  Eclipsing  Binar- 
ies," 241;  theory  of  Cepheid  variables, 
246;  distinction  between  Cepheid  and 
cluster  variables,  246-247;  statistics  of 
eclipsing  binaries,  286.  See  also  Russell 
and  Shapley. 

Solar  spectrum,  description,  28-29. 

Spectral  type  and  color  intensity,  149-151 ; 
effect  on  photographic  image,  145. 

Spectral  type  of  variable  star  and  type  of 
variation,  35-36;  temporary  stars,  35; 
long  period  variables,  35 ;  irregular  vari- 
ables, 36;  short  period  variables,  36; 
Algol  variables,  36. 


Spectrogram,  definition,  219;  measure- 
ment of,  223-225. 

Spectrograph,  Bruce,  description  of,  219- 
223;  testing,  224-225. 

Spectroscope,  18-19. 

Spectroscopic  binary,  definition,  225-226 ; 
connection  with  variable  stars,  227- 
228. 

Spectroscopy,  early  workers  in,  219. 

Spectrum,  formation  of,  17-18;  invisible, 
21;  continuous,  22;  emission,  22;  ab- 
sorption, 23 ;  of  hydrogen,  23 ;  effect  of 
change  in  temperature  and  pressure,  23, 
25-26;  investigations  in,  24-28;  banded, 
25-26;  lined,  26;  enhanced  lines,  26; 
effect  of  pressure  of  magnetic  field, 
26-27;  reversal  of  lines  in,  27;  methods 
of  producing,  27-28;  recent  investiga- 
tions of  hydrogen  and  helium,  37. 

Spectrum  analysis,  laws  of,  22-23;  applica- 
tions, 23-24. 

Stained  plates,  use  of,  in  stellar  pho- 
tography, 142,  148-150. 

Statistical  study  of  variable  stars,  long 
period  variables,  distribution  and  pe- 
riod, 273-275;  color  and  period,  275-278; 
period  and  range,  278-279;  short  period 
variables,  period  and  range,  280;  spec- 
tral types,  281 ;  spectral  type  and  color, 
282;  galactic  distribution,  282-284;  ir- 
regular variables,  range,  color,  spec- 
tral type,  284-285;  eclipsing  binaries, 
286 ;  number  discovered  by  a  single  ob- 
server, 286;  early  discoveries,  286-287; 
conclusions  287-288. 

Stebbins,  selenium  cell,  153,  154, 155-158; 
secondary  minimum  of  Algol,  157;  ele- 
ments of  Algol's  orbit,  230;  surface  in- 
tensity, 238. 

Steinheil,  determination  of  magnitude 
ratio,  p,  96. 

Steinheil  binoculars,  for  observing 
brighter  variables,  64. 

Stellar  spectra,  classification,  Secchi,  30; 
Fraunhofer,  30;  Harvard,  31-34;  devel- 
opment of  types,  34;  description  of 
plates,  315-317. 

Struve,  color  of  double  stars,  136. 

Sutton,  study  of  Cepheid-Geminid  group 
of  variables,  244. 

Telescope,  guiding,  138-139;  penetrating 
power,  299 ;  setting  with  circles,  300-301 ; 
identification  of  field,  301-303. 

Temporary  stars,  description,  6-9;  spec- 
trum, 35;  light  variation,  253;  history, 
253-255. 

Tenth  meter,  21. 

Turner,  306;  interest  in  observations  of 


INDEX 


327 


Pogson,  266;  editor  of  observations  of 
Baxendell,  267;  of  Knott,  267. 
Tycho  Brahe,  globe,  82,  85;Jadopted  Ptol- 
emy's magnitudes,  83;  constellation  of 
Gemini,  84-85. 

Ulugh  Beigh,  magnitudes,  83. 

Upton,  star  atlas,  38, 40. 

Uranometria  Argentina,  62, 63, 91-93, 126, 
306. 

Uranometria  Nova,  88;  magnitude  scale, 
38-39,  92,  93,  126. 

S  UrsaeMajoris,  light  scale  of  comparison 
stars  by  Parkhurst,  180-185 ;  irregulari- 
ties in  elements,  248. 

Valentiner,  editor  of  Schonfeld's  obser- 
vations, 260,  261. 

Van  der  Bilt,  investigation  of  U  Gemino- 
rum,  251. 

Variable  stars,  definition,  3 ;  Pickering's 
classification  of,  5-17 ;  provisional  cata- 
logue of,  5 ;  of  long  period,  9-10 ;  irreg- 
ular, 10-11;  of  short  period,  11-14;  of 
Algol  type,  14-17 ;  cluster  type,  14;  spec- 
tral types,  temporary  stars,  35;  long 
period,  35 ;  irregular,  36 ;  short  period, 
36;  Algol  type,  36;  catalogue  of,  51,  70; 
method  of  Jnaming  variables,  Argelan- 
der's,  68-69;  Harvard,  69;  Chandler's, 
69 ;  Andre's,  69-70';  Nijland,70;  discov- 
ery, 73-80  ;  methods  of  observation,  vis- 
ual, Argelander's  step  method,  103-106; 
Pickering's,  106-107;  other  modifica- 
tions, 107-108  ;  effect  of  color  on  obser- 
vations, 108-109;  precautions,  109-110; 
value  of  photo-electric  photometers  for 
short  period  variables,  157-158, 166;  light 
scale  of  comparison  stars,  170-172;  light 
steps  of  variable,  172-174;  method  of 
plotting  single  light  curve,  174;  deter- 
mination of  time  of  maximum  and 
length  of  period,  174-176 ;  change  from 
light  steps  into  magnitudes,  177-179; 
formation  of  light  scale  with  weights, 
180-185  ;  method  of  deriving  mean  light 
curve,  186-199  ;  its  use,  199-202;  Harvard 
modification  for  long  period  variables, 
202-203;  prediction  of  maximum  and 
minimum,  206-212;  correction  to  geo- 
centric minimum  for  short  period  vari- 


ables, 213-215;  evidence  that  Algol  va- 
riables are  spectroscopic  binaries,  227- 
236;  possible  combinations  in  eclipsing 
systems,  236-240 ;  evidence  for  /3  Lyrae 
type,  241-244;  theories  for  Cepheid- 
Geminid  type,  244-247;  for  cluster  type, 
246-247;  irregularities  in  elements  of 
long  period  variables,  248-252 ;  irregular 
variables,  252-253;  temporary  stars,  253- 
255;  collections  of  observations,  255-270; 
statistical  study,  273-286;  number  of 
discoveries  attributed  to  single  ob- 
server, 286;  earliest  variables  discov- 
ered, 286-287;  value  of  observations  by 
Argelander's  method,  289-294;  method 
of  observation,  294;  amateur  associa- 
tions, 297-298;  identification  of  variable 
in  field,  301-303;  precautions,  303-305; 
list  of  variables  for  beginners,  305-306. 

Variation,  elements  of  stellar,  4-5. 

Velocity  curve,  233-234. 

Vibration  frequency,  21-22,  216. 

Vogel,  early  work  in  spectroscopy,  219; 
spectroscopic  observations  of  Algol, 
229-230,  231;  at  Potsdam,  264. 

Wallace,  stained  plates,  148;  Pan-iso 
plates,  149. 

Webb,  color  of  double  stars,  135. 

Wilson,  value  of  amateur  work  on  vari- 
able stars,  296-297. 

Wing,  Vincent,  Harmonicon  Celeste,  list 
of  stars  in  constellation  of  Gemini,  84- 
85. 

Wolf,  early  photograph  of  Nova  Auri- 
gae,  8;  Palisa'and,  photographic  star 
charts,  50 ;  discovery  of  variables  with 
aid  of  stereo-comparator,  78. 

Yendell,  effect  of  parallactic  angle  on  ob- 
servations, 109-110;  use  of  mean  light 
curve  in  determining  minimum  of  Al- 
gol type,  199-201;  precautions,  304-305. 

Zeeman  effect,  27. 

Zinner,  galactic  distribution  of  variables, 
283-284. 

Zollner  photometer,  116-119;  method  of 
reduction,  119-120;  use  in  Potsdam 
Photometric  Durchmusterung,  120. 


fctoetfibe 

CAMBRIDGE  .  MASSACHUSETTS 
U   .   S   .  A 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


Y   b- 


AFK  6     lUbu 


UBRARY  USE 


LIBRARY  USE 


APR  1ft  1950 


-— rr^- 


a 


LD  21-100m-7,'33 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


